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MATHEMATICS OF COMPUTATIONVOLUME 47. NUMBER 175JULY I4K6. PACIES .161-36«
Computation of All the Amicable Pairs Below 1010
By H. J. J. te Riele
Abstract. An efficient exhaustive numerical search method for amicable pairs is.described.
With the aid of this method all 1427 amicable pairs with smaller member below 1010 have
been computed, more than 800 pairs being new. This extends previous exhaustive work below
108 by H. Cohen. In three appendices (contained in the supplements section of this issue),
various statistics are given, including an ordered list of all the gcd's of the 1427 amicable pairs
below 1010 (which may be useful in further amicable pair research). Suggested by the
numerical results, a theorem of Borho and Hoffmann for constructing APs has been extended.
1. Introduction. Let a(m) denote the sum of all the divisors of m, including 1 and
m. An amicable pair (AP) is a pair of positive integers (m,n), m < n, such that
a(m) = a(n) = m + n. We note that m is abundant (since a(m) > 2m) and that n
is deficient (since a(n) < 2n). The smallest AP is
(220,284) = (225.11,2271).In order to check whether or not a given positive integer m is the smaller member
of an amicable pair, it seems necessary, at first sight, to compute a(m) and
n := a(m) — m, to check whether n > m (i.e., whether m is abundant), and, if so, to
compute a(n) and compare a(m) with a(n). This involves one or two complete
factorizations, in case m is deficient or abundant, respectively. However, a closer
look reveals that it is often possible to find out whether a given number m is
deficient (hence cannot be the smaller member of an AP) without the need to
factorize it completely. Moreover, once o(m) and n (= a(m) - m) have been
computed, it is often possible to discover that a(n) ¥= a(m) without the need to
factorize n completely.
These considerations have guided the design of an efficient exhaustive numerical
AP search algorithm, the details of which are given in Section 2. With the aid of this
algorithm we have extended Cohen's exhaustive list of all 236 APs with smaller
member below 108 [4] to all 1427 APs with smaller member below 1010. Of these,
601 have been published earlier [6], [7]. The other 826 seem to be new, and are
published here for the first time (9 of them have been communicated to the author
already in 1983 and 1984 by Woods (2), Borho (2) and Lee (5)). Section 3 presents
details of the computations together with several tables collected from this search.
Moreover, a result of Borho and Hoffmann for constructing APs is extended, as was
suggested by the numerical tables.
Received March 27, 1985.
1980 Mathematics Subject Classification. Primary 10A40; Secondary 10-04.
Key words and phrases. Amicable pair.
©1986 American Mathematical Society
0025-5718/86 $1.00 + $.25 per page
361
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362 H. J. J TE RIELE
Three appendices to this paper appear in the supplements section of this issue.
These may also be obtained by writing to the author.
In Appendix I, we present the complete list of all 1427 APs with smaller member
below 1010 ordered according to the size of the smaller members of the pairs.
Appendix II displays the same list with a different ordering, viz., according to the
various occurring types (defined in Section 3). Finally, Appendix III tabulates all the
greatest common divisors of the 1427 APs, in increasing order, together with their
frequencies of occurrence, and, for each gcd g, the rank numbers of all the APs
(m, n) for which gcd(w, n) = g.
2. Check Whether a Given m is the Smaller Member of an AP. Let p, be the ;th
prime, P,/= Wk%T1 pk, Qu-= UfJ/1 pk/(pk - 1). We start with the following
lemma which gives an upper bound for a(m)/m.
Lemma 2.1. // m only has prime divisors > p¡ (i > 1) and if m < P¡¡+i (j > 1)
then a(m)/m < Q(j.
Proof. Since m < P¡¡+i = PiP¡+\ ' • • pi+j, and since any prime divisor of m is
> p¡, it follows that m has at most j different prime divisors > p¡ (otherwise we
would have m > p¡p¡+\ ••• pi+j = Pij+i)- This implies that
a(m) n pe+l ~~ i n P ~ P e o P ' rr ^k - n
~™~ = /i p'(p - i) ~ p'I ^P31" < M P7^ * k=\ Jk~^=Q,J' DIn the algorithm below, this lemma is invoked very frequently. Therefore, we require
a precomputed table of P- and g-values, large enough so that the values needed can
be found quickly by simple table look-ups.
Now we describe an efficient algorithm to check whether a given positive integer
m belongs to an AP (m, n) with m < n. This algorithm is based on the observation
that when, for given y and N, we want to verify one of the relations a(N)/N > y,
= y, < y, and when the primes 2,3,...,p have been tried as divisors of N, it may
be possible
(i) to detect, with Lemma 2.1, whether a(N)/N < y by using the information that
the unfactored portion of N only has prime divisors > p, and
(ii) to detect whether a(N)/N > y by using the factored portion of N.
In this way, much unnecessary factorization time may be avoided. The price to pay
for this gain lies in the time needed to consult the P- and Q-tables used in Lemma
2.1. In the algorithm, the index z'max is the maximum value of i for which Lemma 2.1
is invoked. In order to restrict this table look-up time, t'max should not be chosen too
large. The optimal value of z'max also depends on the actual implementation of the
algorithm (cf. Section 3).
Algorithm to Check Whether m is the Smaller Member of an AP.
Step 1. (Find out whether m is abundant; in this step, keep m = mxm2 where
gcd(m1,wi2)= 1, mx is the factored and m2 is the unfactored portion of m,
a := a(mx)/mx; start with mx := 1, m2:= m, a.= 1.)
Start factoring m by trial dividing m2 by the primes px, p2,... < m1/2. In case a
prime power divisor p'_x (e > 1) of m2 has been found, update mx, m2 and a
(mx:= mxpf_x,m2:= m/mx,a:= a • o(pf_x)/pf_x). After the trial division with
p _, (whether or not p¡_x divides m2): if a < 2 and 4 < / < ;'max, check whether m
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AMICABLE PAIRS BELOW 1010 363
is possibly deficient as follows: by inspecting the P-table find the smallest value of j
(='j*) such that m2 < Pjt+i', if «ô,,7» < 2, then STOP (because, in that case, m is
deficient: by Lemma 2.1 we have a(m2)/m2 < Qijt so that
o(m) °(mx) a(m2) a(m2)-=-•-= a-< aQH» < 2).
m mx m2 m2 J
If a > 2, or / < 4 or i > i'max, the deficiency check on m is left out. After the
complete factorization of m (and simultaneous computation of o(m))\ if m < a (m)
— m =.n (i.e., m is abundant), go to Step 2, otherwise STOP.
End of Step 1
Step 2. (Given m, a(m) and n = a(m) - m, check whether a(n) = a(m); during
the factorization of n try to exclude those m for which a(n)¥= a(m) as early as
possible by testing whether a(n)/n *= ß where ß = a(m)/n; in this step, keep
n — nxn2, where gcd(«l5 n2) = 1, nx is the factored and n2 the unfactored portion
of n, a.= a(nx)/nx; start with nx:= 1, n2:= n, a.= 1.)
Start factoring n by trial dividing n2 by the primes px, p2,... < n1/2. In case a
prime power divisor pf_x (e > 1) of n2 has been found, update nx, n2 and a: if the
updated a satisfies a > ß, then STOP (because, in that case, we have
o(n) o(nx) a(n2) a(nx) a(m)-=-^-= a > ß =-,
n nx n2 nx n
so that a(n) =£ a(m)). After the trial division with /?,_, (whether or not p¡_x divides
n2): if 4 < i < ¿max check whether a(n)/n < ß as follows: by inspecting the P-table
find the smallest value of ; (=:/*) such that n2 < Pij+1. If aQijt < ß, then STOP
(because, in that case, a(n)/n < ß: by Lemma 2.1 we have a(n2)/n2 < Q¡ ,, so
that
o(n) o(nx) a(n2) a(n2)——- =-= a-< aQ, ,» < ß ).
n nx n2 n2 *'•' HJ
If / < 4 or i > ;max, the check on a(n)/n < ß is omitted. After the complete
factorization of n (and simultaneous computation of o(n))\ check whether a(n) =
a(m). If so, (m, n) is an AP.
End of Step 2
3. Computing All the APs Below 1010. In order to compute all the APs (m, n) with
m < n and 108 < m < 1010 (thus extending H. Cohen's computations reported in
[4]), we distinguish between m = 0 (mod 6) (the easy case), and m # 0 (mod 6) (the
hard case).
If m = 0 (mod6) and n = a(m) - m is even, then (m, n) cannot be an AP [5].
Therefore, n should be odd. In that case, we have [6] m = 2)íM2, n = N2, with
H e N, M and N being odd. For all the numbers m = 2'"Ai2 with 3|M and
108 < m < 1010, we computed n := a(m) - m and checked whether n was a perfect
square. Not a single such case was found. Computer time was about 6 CPU seconds.
For all m # 0 (mod 6) with 108 < m < 1010 we used the algorithm of Section 2 to
find all APs in this range. The optimal choice of jmax for our FORTRAN-implemen-
tation on a CYBER 750 was about 75. This value was chosen to be fixed for the
whole range. The speed-up factor of our program was about 15, compared with a
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364 H. J. J. TE RIELE
straightforward program which, given m, computes a(m) and, if n:= a(m) - m >
m, computes a(n). A slight increase of the speed was obtained as follows. In Step 1,
in case a prime (power) factor of m2 was found and mx and a(mx) (among others)
were updated, it was checked whether both mx and o(mx) were divisible by one of
the primitive abundant numbers 20 = 225, 28 = 227, 70 = 2.5.7 and 88 = 2311. If
so, the algorithm was stopped since this implied that also m and o(m), hence also
n = a(m) — m were divisible by this abundant number, so that both m and n were
abundant. This is impossible for an AP (m, n).
The total time to cover the range 108 < m < 1010 was about 1000 (low priority)
CPU hours, spent in the last seven months of 1984.
The total number of APs (m,n) found with m < n and 108 < m < 1010 was
1191. In Appendix I (of the supplements section) all the APs with smaller member
< 1010 are given (including the 236 APs with smaller member < 108). For each
pair we list the decimal representation and the prime factorization of the members, a
rank number, a code (letter plus digit) referring to the discoverer, and the type of the
pair (defined below). For example, pair #1427 reads as follows:
1427 9967523980 2E2.257.5.17.37.3083
R942 12890541236 2E2.257.107.117191.
Table 1 gives the meaning of the codes, and their frequencies of occurrence.
Extensive information about the sources of the pairs with code LI is given in the
survey paper [6].
There are 1015 pairs with even members and 412 with odd members. The minimal
and maximal values of m/n are 0.6979 and 0.999858 for the APs #567 and #1010,
respectively.
Let A(x) be the number of APs (m, n) with m < n and m < x. From the list of
APs with m < 108, Bratley et al. [3] concluded that for x < 108, A(x) is approxi-
mately proportional to x1/2/ln(x). In Table 2 we give, for x = k. 109 (1 < k < 10):
A(x), A(x)ln(x)/xl/2, A(x)(\n(x))2/x1/2 and A(x)(\n(x))3/xl/2. From these fig-
ures we may draw the conclusion that for x < 1010, A(x) is approximately propor-
tional to x1/2/(\n(x))3.
Table 1
Status list of the first 1427 APs (m,n), m < n, with m < 1010
code #APs references and remarks
LI
R2Wl
R3
W2R6
W3
L2B4
R9
5081
73
19
11
1
52
816
[6]
[9] (#1056)sent to the author by D. Woods on June 29,1982 and published in
[7]found by the author with the methods described in [8], and
published in [7]
sent in by D. Woods on Feb. 16,1983 (#330)
found by the author in May, 1983 (#1375)
sent in by D. Woods on My 11,1983 (#1050)
sent in by E. J. Lee in July, 1984 (# #778,860,894,1241,1261)sent in by W. Borho on Nov. 2,1984 (# #809,1393)
found by the author during the systematic search described in this
paper
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AMICABLE PAIRS BELOW 1010 365
Table 2
Comparison of A(x) with x1/2/(\n(x))', i = 1,2,3
A'/IO" A(x) A(x)\n(x)/x^ A(,x)(ln(x))2/x1/2 A(x)<\vt(x))i/x 1/2
1
2
34
56
7
S
9
10
586762
8981009
1100118512561317
13771427
0.38400.3649
0.35780.3527
0.34740.3444
0.34030.33580.33270.3286
7.9587.8157.8077.799
7.7597.7557.7157.6567.6257.566
164.9167.4
170.4172.4
173.3174.6
174.9174.6
174.8174.2
We define an AP (m, n), m < n, to be a regular amicable pair of type (i, j), if
(m, n) = (gM, gN), where g = gcd(w, n), gcd(g, M) = gcd(g, N) = 1, M and N
are squarefree, and the numbers of prime factors of M and TV are i and j,
respectively. Other pairs are called irregular or exotic. There are 1082 regular and
345 irregular APs with smaller member < 1010. It is easy to see that there are no
regular pairs of type (1, j), j > 1: let g be the gcd of such an AP, so that
(m,n) = (gp,gN) where p is a prime and gcd(g, p) = gcd(g,N) = 1. We have
m < n, hence p < N. By definition, a(gp) = o(gN), implying that p + 1 = a(N).
Since, for any N e M, a(N)> N, this implies that p + 1 > N, a contradiction. We
note that in this argument N need not be squarefree.
In Table 3 we give the frequency distribution of the various types among the first
1082 regular APs. We note that there are relatively few regular APs of type (j, 1),
/' > 2, and of type (/', j) with /' < j.
In [7] the total number of known APs with smaller member < 1010 was 601
(these are the APs belonging to the first four codes in Table 1). Among them were
104 irregular APs, i.e., 17.3%. Comparing this figure with the 345 irregular APs in
our complete list of APs with smaller member < 1010, i.e., 24.2%, we see that
relatively many irregular APs were found in our systematic search.
In Appendix II (of the supplements section) we present lists of all the 1082 regular
APs arranged according to their types, together with a list of the 345 exotic APs.
This appendix may be useful for searches of APs of a special type.
The regular pairs of type (;', 1), /' > 2, play an important role as "mother" pairs in
methods to generate new APs from given pairs. In [8] a substantial part of the new
APs found there was constructed from such mother pairs. In [1], Borho and
Hoffmann have partially generalized the methods from [8] by introducing the
concept of a breeder: a breeder is a pair of positive integers (ax, a2) such that the
equations
ax + a2x = a(ax) = a(a2)(x + 1)
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366 H. J. J. TE RIELE
Table 3
Frequency distribution of the first 1082 regular APs
of type (i,j), i > 2, j> 1
column
totals
3
20
16
1
0
67
271
78
6
21
280201
IX
4
24
63
7
37 422 520 98
row totals
112
591
345
34
1082
have a positive integer solution x. If x is a prime, then (ax, a2x) is an amicable pair.
For certain breeders, called "special" breeders, Borho and Hoffmann formulate the
following
Theorem 1 [1]. Let (ax,a2) be a special breeder, i.e., ax = au, a2 = a, with
gcd(a, u) = 1. Take any factorization of C:= a(u)(u + o(u) - 1) into two different
factors Dx, D2 (C = DXD2). Then, if the numbers s¡ = D¡ + a(u) - 1, for i = 1, 2,
and also q = u + sx + s2 are primes not dividing a, then (auq, asxs2) is an amicable
pair. D
Regular APs of type (/, 1), i > 2, are of the form (au,ap), p prime, and the
numbers (au, a) are special breeders which generally produce many APs with the
above theorem.
In our list of 1427 APs we found a few APs, e.g., #647 and #955, which
suggested that the condition gcà(a, u) = 1 in Theorem 1 may be dropped. In fact,
we have
Theorem 2. Let (au, a) be a breeder, i.e., there exists a positive integer x such that
au + ax = o(au) = a(a)(x + 1). Take any factorization ofC:= (x + l)(x + u) into
two different factors Dx, D2(C = DXD2). Then, if the numbers 5, = D¡ + x, for i = 1,
2, and also q = u + sx + s2 are primes not dividing a, then (auq, asxs2) is an
amicable pair. D
The proof of this theorem is left to the reader.
If gcd(a, u) = 1, then a(au) = a(a)a(u), so that x = a(u) - 1 and Theorem 2
reduces to Theorem 1. As an example, AP #955 gives the breeder (au, a) with
a = 3.5.7.19 and u = 7.29.47.181. Theorem 2 yields 16 new APs with this breeder as
input.
It is known [5] that most even APs have a pair sum which is = 0 (mod 9). Our
search proves that indeed Poulet's pair #503: (24331.19.6619,24331.199.661) is the
smallest exceptional pair. All known exceptional pairs had members = 7 (mod 9)
and a pair sum = 5 (mod 9). In our search, we found two even APs with pair sum
= 3 (mod 9), viz., the (irregular) pairs:
ÄV77.94/l92103.1627 and #874.22i9/l3237.43.139*577' 2 \3847.16763 ^ #™' " 41.151.6709.
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AMICABLE pairs BELOW 10 367
Table 4
The 17 APs among the first 1427', whose pair sum is ^ 0 (mod 9)
regular
irregular
even members odd members
#503, type (2,2)#1031, type (2,2)
#1081, type (2,2)
#899, type (3,2)#1057, type (2,2)
#1158, type (3,2)
##577,874 ##7,38,78,113,256,440,1083,1175,1380
Table 5
All (37) pairs from the first 1427 APs having the same pair sum
rank numbers pair sum
prime decomposition of the pair sum,
i.e., exponents belonging to the primes
2 3 5 7 11 13 17 19 23 29 31 37
32 35105 109
137 138172 173272 276
282 286350 351347 355373 375
368 377395 399411 415
427 433462 476486 491
574 582626 630
653 665695 697717 730751 753
798 807786 787
824 840940 941926 952
997 998
1012 1019
1069 10971124 1142
1147 1150
1143 11811232 12331254 1265
1249 12551272 1278
1410 1425
129600020528640
3773952075479040
321408000
348364800556839360579156480638668800
661893120761177088796340160
8838720001181174400
1282417920
20684160002395008000
2682408960
315502387235997696004049740800
4606156800
4716601344
509483520068245632006897623040
79252992008273664000
1002792960011195712000
114162048001209821184013473008640
14341017600
14478912000
1505806848019926466560
7
9
10
31
1 1
11 4 1 110 4
13 56 6
912
12
10
6
8
7
8
9
10
12
11
13
10
13
13
109
13
11
11
3
2 11 1
5 1 24 2 1
5 1 1
1
1 2
3
2 21 13 1
27
37
5
512 59 39
12
10
12
9
10
6 2 1
3 2 2
1 1
2 2
3 1232 1
11
2
3
1
14 5 1 1 1
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368 H. J. J. TE RIELE
These are the first two examples of APs of the form described in [5, Theorem I, case
(b)] (also cf. the remarks immediately following Table I in [5]). Table 4 gives the
rank numbers of the 17 APs with smaller member < 1010 whose pair sum is # 0
(mod 9), divided into even and odd pairs, and regular and irregular pairs.
Another question, suggested by Professor C. Pomerance, is whether pairs, triples,
quadruples, etc. of APs exist having the same pair sum. Among the first 1427 APs,
we found 37 such pairs of APs, but no such triples, quadruples, etc. Table 5 gives the
rank numbers of these pairs of APs, and the prime factorization of their pair sums.
The pair sums only have prime divisors < 37. In 30 of the 37 cases at least one
member of the pair was found during the exhaustive search described in the present
paper.
In Appendix III (of the supplements section) we tabulate all the greatest common
divisors of the first 1427 APs, ordered according to their size, with frequencies, and
with the rank numbers of all the APs corresponding to a given gcd. This might be
useful in further searches for special APs, and in searches for so-called isotopic APs
(cf., [6, p. 83]). For example, new APs, isotopic with APs from the list of 1427 APs,
are obtained by replacing the common factor 335 in # #882 and 1087 by 327.13, by
replacing the common factor 3353 in #1205 by 325231, and by replacing the
common factor 335231 in ##717 and 1228 by 365.23.137.547.1093, and by
3105.23.107.3851.
In [8], we have presented methods to find new APs from known APs. By applying
these methods to the new APs among the first 1427 APs, we have found 117 new
APs (with smaller member > 1010). The new APs were found mainly from mother
pairs having a relatively simple structure, like those of type (/', 1), i > 1. They will be
published in a forthcoming report [2], together with many other new amicable pairs.
Centre for Mathematics and Computer Science
Kruislaan 413
1098 SJ Amsterdam
The Netherlands
1. W. Borho & H. Hoffmann, "Breeding amicable numbers in abundance," Math. Comp., v. 46, 1986,
pp. 281-293.2. W. Borho, H. Hoffmann & H. J. J. te Riele, Table of Amicable Pairs Between 1010 and 1052,
CWI-report. (In preparation.)
3. P. Bratley, F. Lunnon & John McKay, "Amicable numbers and their distribution," Math. Comp.,
v. 24,1970, pp. 431-432.4. H. Cohen, "On amicable and sociable numbers," Math. Comp., v. 24, 1970, pp. 423-429.
5. E. J. Lee, " On divisibility by nine of the sums of even amicable pairs," Math. Comp., v. 23,1969, pp.
545-548.6. E. J. Lee & J. S. Madachy, "The history and discovery of amicable numbers," J. Recreational
Math., v. 5, 1972; Part I: pp. 77-93, Part II: pp. 153-173, Part III: pp. 231-249.
7. H. J. J. TE RlELE, Table of 1869 New Amicable Pairs Generated from 1575 Mother Pairs, Report NN
27/82, Math. Centre, Amsterdam, Oct. 1982.
8. H. J. J. te Riele, "On generating new amicable pairs from given amicable pairs," Math. Comp., v.
42, 1984, pp. 219-223.9. H. J. J. TE RlELE, Further Results on Unitary Aliquot Sequences, Report NW 2/78, Math. Centre,
Amsterdam, 2nd ed., Jan. 1978.
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MATHEMATICS OF COMPUTATIONVOLUME 47. NUMBER 175JULY \1Xb. I'A(iESS9-S40
Supplement to
Computation of All the Amicable Pairs Below 1010
By H. J. J. te Riele
Appendix I
The first 1427 APs
1 220 2E2.5.11 3DLI 21 284 2E2.71
2 1184 2E5.37LI X 1210 2.5.11E2
3 2620 2E2.5.131LI 22 2924 2E2.17.43
4 5020 2E2.5.251LI 22 5564 2E2.13.107
5 6232 2E3.19.41LI X 6368 2E5.199
6 10744 2E3.17.79LI 22 10856 2E3.23.59
7 12285 3E3.5.7.13LI X 14595 3.5.7.139
8 17296 2E4.23.47LI 21 18416 2E4.1151
9 63020 2E2.23.5.137LI 21 76084 2E2.23.827
10 66928 2E4.47.89LI 22 66992 2E4.53.79
11 67095 3E3.5.7.71LI 22 71145 3E3.5.17.31
12 69615 3E2.7.13.5.17LI 21 87633 3E2.7.13.107
13 79750 2.5E3.11.29LI X 88730 2.5.19.467
14 100485 3E2.5.7.11.29LI 32 124155 3E2.S.31.89
15 122265 3E2.5.13.11.19LI 21 139815 3E2.5.13.239
16 122368 2E9.239LI X 123152 2E4.43.179
17 141664 2E5.19.233LI X 153176 2E3.41.467
18 142310 2.5.7.19.107LI 32 168730 2.5.47.359
19 171856 2E4.23.467LI 22 176336 2E4.103.107
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22 196724 2E2.11.17.263LI 22 202444 2E2.11.43.107
23 280540 2E2.5.13E2.83LI X 365084 2E2.187.853
24 308620 2E2.5.13.1187LI 32 389924 2E2.43.2267
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25 319550LI X 430402
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27 437456 2E4.19.1439LI 22 455344 2E4.149.191
28 469028 2E2.7E2.2393LI X 486178 2.7E2.11E2.41
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30 522405 3E2.5.13.19.47LI 22 525915 3E2.5.13.29.31
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32 609928 2E3.11.29.239LI 32 686072 2E3.191.449
33 624184 2E3.11.41.173LI 32 691256 2E3.71.1217
34 635624 2E3.11.31.233LI 32 712216 2E3.127.701
35 643336 2E3.29.47.59LI 32 652664 2E3.17.4799
36 667964 2E2.11.17.19.47LI 43 783556 2E2.31.71.89
37 726104 2E3.17.19.281LI 32 796696 2E3.53.1879
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39 879712 2E5.37.743LI X 901424 2E4.53.1063
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41 947835 3E3.5.7.17.59LI 32 1125765 3E3.5.31.269
42 998104 2E3.17.41.179LI 32 1043096 2E3.23.5669
43 1077890 2.5.11.41.239LI 33 1099390
44 1154450LI 22 1189150
45 1156870LI 32 1292570
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46 1175265 3E2.7E2.13.5.41LI 21 1438983 3E2.7E2.13.251
47 1185376 2E5.17.2179LI X 1286744 2E3.41.3923
48 1280565 3E2.5.13.11.199LI 22 1340235 3E2.5.13.29.79
49 1328470 2.5.11.13.929LI X 1483850 2.5E2.59.503
50 1358595 3E2.5.19.7.227LI 22 1486845 3E2.5.19.37.47
51 1392368 2E4.17.5119LI 22 1464592 2E4.239.383
52 1466150 2.5E2.7.59.71LI X 1747930 2.5.47.3719
53 1468324 2E2.11.13.17.151LI 43 1749212 2E2.37.53.223
54 1511930 2.5.7.21599LI 23 1598470 2.5.19.47.179
55 1669910 2.5.11.17.19.47LI 42 2062570 2.5.239.863
56 1798875 3E3.5E3.13.41LI X 1870245 3E2.5.13.23.139
57 2082464 2E5.59.1193LI 22 2090656 2E5.79.827
58 2236570 2.5.7.89.359LI 33 2429030 2.5.23.59.179
59 2652728 2E3.13.23.1109LI 32 2941672 2E3.71.5179
60 2723792 2E4.37.43.107LI 32 2874064 2E4.263.683
©1986 American Mathematical Society
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SUPPLEMENT S15
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SUPPLEMENT SI 7
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SI 8 SUPPLEMENT
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SUPPLEMENT SI 9
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S20 SUPPLEMENT
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SUPPLEMENT S21
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Appendix II
The first 1427 APs ordered according
to the various occurring types
AMICABLE PAIRS OF TYPE {2,1)
LI 218
LI 219
LI 2112
LI 2115
LI 2146
LI 21104
LI 21117
LI 21
162LI 21
291LI 21
297LI 21
303LI 21
460LI 21
629LI 21
640LI 21
792LI 21
888LI 21
1030LI 21
1191LI 21
1219LI 21
220 2E2.5.11284 2E2.7117296 2E4.23.4718416 2E4.115163020 2E2.23.5.13776084 2E2.23.82769615 3E2.7.13.5.1787633 3E2.7.13.107122265 3E2.5.13.11.19139815 3E2.5.13.2391175265 3E2.7E2.13.5.411438983 3E2.7E2.13.2519363584 2E7.191.3839437056 2E7.7372711498355 3E4.5.11.29.S912024045 3E4.5.11.269931536855 3E2.5.7.53.188932148585 3E2.5.7.102059175032884 2E2 . 13.17.389.509175826716 2E2.13.17.198899183408615 3E2.5.13.19.29.569190055385 3E2.5.13.19.17099196421715 3E2.5.19.37.7.887224703405 3E2.5.19.37.7103536637465 3E2.7E2.13.97.5.193646745463 3E2.7E2.13.97.11631191953763 3E2.7E2.11.13.41.4611223611389 3E2.7E2.11.13.194031225052829 3E4.7.11.29.13.5211321639011 3E4.7.11.29.73072172649216 2E8.257.330232181168896 2EB.85201912935281375 3E3 . 5E3.13.149.4492961518625 3E3.5E3.13.674994149106335 3E4.5.11E3.43.1794268776545 3E4.5.11E3.79196066248175 3E2 . 5E2 . 13 . 31 .149 . 4496120471825 3E2.5E2.13.31.674996370495978 2.7E2.19.23.11.135236950103062 2.7E2.19.23.162287
AMICABLE PAIRS OF TYPE (3,1):
86 6955216 2E4.19.137.167LI 31 7418864 2E4.463679
151 23358248 2E3.37.23.47.73LI 31 25233112 2E3.37.85247
164 32205616 2E4.17.167.709LI 31 34352624 2E4.2147039
196 52695376 2E4.17.151.1283LI 31 56208368 2E4.3513023
270 147366765 3E2.7E2.13.5.53.97LI 31 182028483 3E2.7E2.13.31751
312 205843365 3E2.7E2.13.5.43.167LI 31 254264283 3E2.7E2.13.44351
390 347263216 2E4.17.137.9319LI 31 370414064 2E4.23150879
446 492275992 2E3.131.13.23.1571R9 31 553544168 2E3.131.528191
648 1254255550 2.5E2.23.19.137.419R9 31 1333078850 2. 5E2.23.1159199
661 1309651310 2.5.11.29.571.719R9 31 1359071890 2.5.11.12355199
753 1957374968 2E3.31.17.107.4339LI 31 2092365832 2E3.31.8436959
782 2115211995 3E3.5.13.17.31.2287R9 31 2312891685 3E3.5.13.1317887
979 3693013664 2E5.41.131.21487LI 31 3812143072 2E5.119129471
1009 3986534090 2.5.929.7.11.5573LI 31 4971106870 2.5.929.535103
1228 6562770525 3E3.5E2.31.17.19.971R9 31 7322055075 3E3.5E2.31.349919
1300 7696871576 2E3.19.53.127.7523R9 31 7904894824 2E3.19.52005887
TOTAL NUMBER: 16
TOTAL NUMBER: 20
AMICABLE PAIRS OF TYPE (4,1):
779 2099442345 3.5.7.11.13.37.3779LI 41 2533809495 3.5.7.24131519
TOTAL NUMBER:
S22
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SUPPLEMENT S23
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Appendix III
The gcd's of the first 1427 APs
GCD PREQ RANK NUMBER(S) OF AP'S WITH THIS GCD
1 3 4 23 24 36 53 64 83 97115 153 154 165 209 226 238 255 267 274294 313 323 345 347 348 502 505 570 586632 645 721 738 748 750 763 776 800 807811 822 842 846 852 867 877 909 920 937975 976 1060 1119 1132 1160 1251 1307 1316 1322
1350 1355 1365 1368 1374 1399 1406
37 40 42 47 59 62 65 67 69 7476 80 81 84 90 93 94 95 103 110
112 116 120 122 126 131 140 143 144 148150 159 161 163 174 180 182 184 186 192199 201 206 208 212 217 218 239 240 244258 266 279 280 293 296 298 300 305 311314 315 317 318 330 333 339 353 357 361363 374 375 376 382 387 400 405 417 432438 447 452 455 458 459 463 467 475 498513 523 525 546 552 553 558 571 574 579580 587 591 599 606 631 633 639 644 646652 658 665 668 690 702 713 716 720 722727 737 762 771 788 806 808 814 836 845848 854 855 858 863 869 871 876 881 912913 919 925 942 946 953 972 973 974 978981 982 992 999 1011 1026 1028 1043 1063 1077
1078 1079 1113 1117 1120 1126 1130 1141 1152 11531164 1170 1182 1184 1213 1218 1222 1235 1243 12501252 1253 1279 1302 1320 1329 1345 1354 1356 13611376 1378 1391 1393 1404 1407 1408 1417
13 18 43 45 49 52 54 55 58 7075 79 98 99 105 107 128 146 190 210
219 227 228 264 275 283 286 289 299 328332 343 378 420 437 441 448 476 479 507519 534 544 589 598 609 617 620 627 664670 674 676 723 759 769 826 834 835 843859 897 904 1001 1007 1023 1027 1042 1053 1062
1064 1066 1074 1093 1103 1104 1105 1108 1111 11231148 1159 1167 1209 1221 1240 1257 1266 1285 12871373 1394 1398 1419
25 61 73 77 119 127 135 177 187 198249 377 395 399 412 470 480 651 701 756790 809 825 857 873 1070 1110 1342 1344 1384
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