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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.

%I A359334 #83 Feb 16 2025 08:34:04
%S A359334 67212,1296000,20528640,37739520,75479040,321408000,348364800,
%T A359334 556839360,579156480,638668800,661893120,761177088,796340160,
%U A359334 883872000,1181174400,1282417920,2068416000,2395008000,2682408960,3155023872,3599769600,4049740800,4606156800,4716601344
%N 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.
%C A359334 From _Michel Marcus_, Dec 31 2022: (Start)
%C A359334 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.
%C A359334 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.
%C A359334 The other terms correspond to values appearing at least twice in A180164.
%C A359334 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].
%C A359334 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)
%D A359334 Song Y. Yan, Perfect, Amicable and Sociable Numbers, World Scientific Pub Co Inc, 1996, pp. 113-121.
%H A359334 Michel Marcus, <a href="/A359334/b359334.txt">Table of n, a(n) for n = 1..233</a>
%H A359334 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AmicablePair.html">Amicable Pair</a>
%e A359334 67212 is a term because 67212 = 220 + 66992 = 284 + 66928 where (220, 284) and (66928, 66992) are two amicable pairs.
%e A359334 1296000 is a term because 1296000 = 609928 + 686072 = 643336 + 652664 where (609928, 686072) and (643336, 652664) are two amicable pairs.
%Y A359334 Cf. A002025, A063990, A259180, A259933, A036471, A180164, A001065, A066539.
%K A359334 nonn
%O A359334 1,1
%A A359334 _Zoltan Galantai_, Dec 26 2022
%E A359334 More terms from _Amiram Eldar_, Dec 31 2022