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.

0, 1, 2, 4, 7, 8, 16, 32, 39, 42, 64, 128, 175, 256, 291, 292, 512, 537, 1024, 2048, 2087, 2090, 2181, 2184, 2350, 4096, 8192, 8267, 16384, 16437, 16902, 16912, 32768, 34983, 34986, 65536, 131072, 131342, 131363, 131364, 133127, 133130, 133152, 262144, 524288

Offset: 1

Author

Rémy Sigrist, Sep 26 2020

Keywords

• nonn
• base

Comments

In other words, this sequence corresponds to the number m such that A271410(m) divides A029931(m).

For any n > 0, A125297(n) gives the number of positive terms < 2^n.

Every power of 2 belongs to the sequence.

Examples

			The first terms, alongside their binary representation and corresponding set S, are:
  n   a(n)  bin(a(n))   S
  --  ----  ----------  ------------------
   1     0           0  {}
   2     1           1  {1}
   3     2          10  {2}
   4     4         100  {3}
   5     7         111  {1, 2, 3}
   6     8        1000  {4}
   7    16       10000  {5}
   8    32      100000  {6}
   9    39      100111  {1, 2, 3, 6}
  10    42      101010  {2, 4, 6}
  11    64     1000000  {7}
  12   128    10000000  {8}
  13   175    10101111  {1, 2, 3, 4, 6, 8}
  14   256   100000000  {9}
  15   291   100100011  {1, 2, 6, 9}
  16   292   100100100  {3, 6, 9}
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..316
• Mathematics Stack Exchange, A finite set of distinct positive numbers is special if each integer in the set divides the sum of all integers within the set.
• Index entries for sequences related to binary expansion of n

Crossrefs

Cf. A029931, A125297, A271410.

Programs

PARI

is(n) = { my (b=Vecrev(binary(n)), s=select(k -> b[k], [1..#b])); vecsum(s) % lcm(s)==0 }