A180164 -id:A180164 - cp's OEIS frontend

A291550 Even amicable pairs whose sum is not divisible by 9.

666030256, 696630544, 967947856, 1031796176, 2840437756, 3156691844, 4150593232, 4213181968, 4796703664, 4855069456, 10168398448, 11159012552, 33707179456, 33844856128, 53151801712, 55270703248, 62393407792, 65270990608, 107122129216, 107620508608, 284624443948, 287125556692

Offset: 1

Michel Marcus, Aug 26 2017

Most even amicable pairs have a sum which is divisible by 9; for instance, the sum of (220, 284) is 504.

			(666030256, 696630544) is an amicable pair whose sum is 571750000640 == 8 (mod 9).

Amiram Eldar, Table of n, a(n) for n = 1..1134 (terms below 10^20)
Elvin Lee, On divisibility by nine of the sums of even amicable pairs, Math. Comp. 23 (1969), 545-548.

Cf. A063990, A180164, A259180, A291422.

A359334 Amicable numbers k that can be expressed as a sum k = x+y = A001065(x) + A001065(y) and a sum k = z+t = A001065(z) + A001065(t) where (x, y, z, t) are parts of two amicable pairs and A001065(i) is the sum of the aliquot parts of i.

Original entry on oeis.org

67212, 1296000, 20528640, 37739520, 75479040, 321408000, 348364800, 556839360, 579156480, 638668800, 661893120, 761177088, 796340160, 883872000, 1181174400, 1282417920, 2068416000, 2395008000, 2682408960, 3155023872, 3599769600, 4049740800, 4606156800, 4716601344

Offset: 1

Zoltan Galantai, Dec 26 2022

nonn

From Michel Marcus, Dec 31 2022: (Start)
In other words, numbers k that can be expressed as a sum k = x+y = z+t where either (x,y) and (z,t), or (x,z) and (y,t), are 2 amicable pairs.
Note that there is currently a single instance of the case (x,z) and (y,t), and this corresponds to the value 64 that appears twice in A066539.
The other terms correspond to values appearing at least twice in A180164.
There are instances where it can appear 3 times, and the least instance is 64795852800 for the 3 amicable pairs [29912035725, 34883817075], [31695652275, 33100200525], [32129958525, 32665894275].
There are instances where it can appear 6 times, and the least instance is 4169926656000 for the 6 amicable pairs [1953433861918, 2216492794082], [1968039941816, 2201886714184], [1981957651366, 2187969004634], [1993501042130, 2176425613870], [2046897812505, 2123028843495], [2068113162038, 2101813493962]. (End)

			67212 is a term because 67212 = 220 + 66992 = 284 + 66928 where (220, 284) and (66928, 66992) are two amicable pairs.
1296000 is a term because 1296000 = 609928 + 686072 = 643336 + 652664 where (609928, 686072) and (643336, 652664) are two amicable pairs.

Song Y. Yan, Perfect, Amicable and Sociable Numbers, World Scientific Pub Co Inc, 1996, pp. 113-121.

Michel Marcus, Table of n, a(n) for n = 1..233
Eric Weisstein's World of Mathematics, Amicable Pair

Cf. A002025, A063990, A259180, A259933, A036471, A180164, A001065, A066539.

More terms from Amiram Eldar, Dec 31 2022

A229953 Numbers k for which k = sigma(sigma(x)) = sigma(sigma(y)) for some x and y such that k = x + y.

Original entry on oeis.org

4, 8, 32, 60, 128, 8192, 43200, 69360, 120960, 131072, 524288, 4146912, 6549984, 12927600, 13335840, 16329600, 34715520, 51603840, 57879360, 59633280, 107775360, 160797000, 169155840, 252067200, 371226240, 391789440, 436230144, 439883136, 489888000, 657296640

Offset: 1

Paolo P. Lava, Oct 04 2013

nonn

A072868 is a subsequence of this sequence. Any term x of A072868 can be expressed as x = 2*sigma(sigma(x/2)).
Note the analogy with amicable pair sums (A180164) which satisfy a similar condition: k = sigma(x) = sigma(y) where k = x + y. - Michel Marcus, Oct 07 2013
When terms do not belong to A072868, then they belong to A159886, and the (x,y) couples are (23,37), (14999,28201), (34673,34687), (55373,65587), (2056961,2089951), (2458187,4091797), (4586987,8340613), (5174363,8161477), (6204767,10124833), (15788453,18927067), (25748273,25855567), (20699927,37179433), (22239647,37393633), ... - Michel Marcus, Oct 08 2013

			4 = 2 + 2 = 2*sigma(sigma(2)).
8 = 4 + 4 = 2*sigma(sigma(4)).
32 = 16 + 16 = 2*sigma(sigma(16)).
60 = 23 + 37 = sigma(sigma(23)) = sigma(sigma(37)).
128 = 64 + 64 = 2*sigma(sigma(64)).
8192 = 4096 + 4096 = 2*sigma(sigma(4096)).

Graeme L. Cohen and Herman J. J. te Riele, Iterating the sum-of-divisors function, Experiment. Math. (1996) vol. 5, no. 2, pp. 91-100 (see merge at n=60 in tree of section 4 page 97).

Cf. A000203, A072868.

with(numtheory); P:=proc(q) local j,n;
for n from 1 to q do for j from 1 to trunc(n/2) do
if sigma(sigma(j))=sigma(sigma(n-j)) and sigma(sigma(j))=n then print(n);
fi; od; od; end: P(10^6);

a(7)-a(20) from Giovanni Resta, Oct 06 2013

a(21)-a(30) from Donovan Johnson, Oct 08 2013

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A291550 Even amicable pairs whose sum is not divisible by 9.

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A359334 Amicable numbers k that can be expressed as a sum k = x+y = A001065(x) + A001065(y) and a sum k = z+t = A001065(z) + A001065(t) where (x, y, z, t) are parts of two amicable pairs and A001065(i) is the sum of the aliquot parts of i.

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A229953 Numbers k for which k = sigma(sigma(x)) = sigma(sigma(y)) for some x and y such that k = x + y.

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