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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.

%I A229953 #30 May 09 2021 22:12:17
%S A229953 4,8,32,60,128,8192,43200,69360,120960,131072,524288,4146912,6549984,
%T A229953 12927600,13335840,16329600,34715520,51603840,57879360,59633280,
%U A229953 107775360,160797000,169155840,252067200,371226240,391789440,436230144,439883136,489888000,657296640
%N A229953 Numbers k for which k = sigma(sigma(x)) = sigma(sigma(y)) for some x and y such that k = x + y.
%C A229953 A072868 is a subsequence of this sequence. Any term x of A072868 can be expressed as x = 2*sigma(sigma(x/2)).
%C A229953 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 A229953 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 A229953 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 A229953 4 = 2 + 2 = 2*sigma(sigma(2)).
%e A229953 8 = 4 + 4 = 2*sigma(sigma(4)).
%e A229953 32 = 16 + 16 = 2*sigma(sigma(16)).
%e A229953 60 = 23 + 37 = sigma(sigma(23)) = sigma(sigma(37)).
%e A229953 128 = 64 + 64 = 2*sigma(sigma(64)).
%e A229953 8192 = 4096 + 4096 = 2*sigma(sigma(4096)).
%p A229953 with(numtheory); P:=proc(q) local j,n;
%p A229953 for n from 1 to q do for j from 1 to trunc(n/2) do
%p A229953 if sigma(sigma(j))=sigma(sigma(n-j)) and sigma(sigma(j))=n then print(n);
%p A229953 fi; od; od; end: P(10^6);
%Y A229953 Cf. A000203, A072868.
%K A229953 nonn
%O A229953 1,1
%A A229953 _Paolo P. Lava_, Oct 04 2013
%E A229953 a(7)-a(20) from _Giovanni Resta_, Oct 06 2013
%E A229953 a(21)-a(30) from _Donovan Johnson_, Oct 08 2013