cp's OEIS Frontend

A337744 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.

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