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%I #30 May 09 2021 22:12:17
%S 4,8,32,60,128,8192,43200,69360,120960,131072,524288,4146912,6549984,
%T 12927600,13335840,16329600,34715520,51603840,57879360,59633280,
%U 107775360,160797000,169155840,252067200,371226240,391789440,436230144,439883136,489888000,657296640
%N Numbers k for which k = sigma(sigma(x)) = sigma(sigma(y)) for some x and y such that k = x + y.
%C A072868 is a subsequence of this sequence. Any term x of A072868 can be expressed as x = 2*sigma(sigma(x/2)).
%C 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
%C 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
%H Graeme L. Cohen and Herman J. J. te Riele, <a href="http://projecteuclid.org/euclid.em/1047565640">Iterating the sum-of-divisors function</a>, Experiment. Math. (1996) vol. 5, no. 2, pp. 91-100 (see merge at n=60 in tree of section 4 page 97).
%e 4 = 2 + 2 = 2*sigma(sigma(2)).
%e 8 = 4 + 4 = 2*sigma(sigma(4)).
%e 32 = 16 + 16 = 2*sigma(sigma(16)).
%e 60 = 23 + 37 = sigma(sigma(23)) = sigma(sigma(37)).
%e 128 = 64 + 64 = 2*sigma(sigma(64)).
%e 8192 = 4096 + 4096 = 2*sigma(sigma(4096)).
%p with(numtheory); P:=proc(q) local j,n;
%p for n from 1 to q do for j from 1 to trunc(n/2) do
%p if sigma(sigma(j))=sigma(sigma(n-j)) and sigma(sigma(j))=n then print(n);
%p fi; od; od; end: P(10^6);
%Y Cf. A000203, A072868.
%K nonn
%O 1,1
%A _Paolo P. Lava_, Oct 04 2013
%E a(7)-a(20) from _Giovanni Resta_, Oct 06 2013
%E a(21)-a(30) from _Donovan Johnson_, Oct 08 2013
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