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%I #21 Aug 12 2022 19:19:57
%S 0,1,2,4,7,8,16,32,39,42,64,128,175,256,291,292,512,537,1024,2048,
%T 2087,2090,2181,2184,2350,4096,8192,8267,16384,16437,16902,16912,
%U 32768,34983,34986,65536,131072,131342,131363,131364,133127,133130,133152,262144,524288
%N Numbers of the form Sum_{e in S} 2^(e-1) where S is a finite set of positive integers such that any element of S divides the sum of the elements of S.
%C In other words, this sequence corresponds to the number m such that A271410(m) divides A029931(m).
%C For any n > 0, A125297(n) gives the number of positive terms < 2^n.
%C Every power of 2 belongs to the sequence.
%H Rémy Sigrist, <a href="/A337744/b337744.txt">Table of n, a(n) for n = 1..316</a>
%H Mathematics Stack Exchange, <a href="https://math.stackexchange.com/questions/3779929/a-finite-set-of-distinct-positive-numbers-is-special-if-each-integer-in-the-set">A finite set of distinct positive numbers is special if each integer in the set divides the sum of all integers within the set.</a>
%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%e The first terms, alongside their binary representation and corresponding set S, are:
%e n a(n) bin(a(n)) S
%e -- ---- ---------- ------------------
%e 1 0 0 {}
%e 2 1 1 {1}
%e 3 2 10 {2}
%e 4 4 100 {3}
%e 5 7 111 {1, 2, 3}
%e 6 8 1000 {4}
%e 7 16 10000 {5}
%e 8 32 100000 {6}
%e 9 39 100111 {1, 2, 3, 6}
%e 10 42 101010 {2, 4, 6}
%e 11 64 1000000 {7}
%e 12 128 10000000 {8}
%e 13 175 10101111 {1, 2, 3, 4, 6, 8}
%e 14 256 100000000 {9}
%e 15 291 100100011 {1, 2, 6, 9}
%e 16 292 100100100 {3, 6, 9}
%o (PARI) is(n) = { my (b=Vecrev(binary(n)), s=select(k -> b[k], [1..#b])); vecsum(s) % lcm(s)==0 }
%Y Cf. A029931, A125297, A271410.
%K nonn,base
%O 1,3
%A _Rémy Sigrist_, Sep 26 2020
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