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

1,3

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.

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

Example

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}

Prog

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

Crossrefs

Cf. A029931, A125297, A271410.

Sequence in context: A327099 A338888 A369703 * A045594 A013056 A013131

Adjacent sequences: A337741 A337742 A337743 * A337745 A337746 A337747

Keyword

nonn,base

Author

Rémy Sigrist, Sep 26 2020

Status

approved