A175381 A positive integer k is included if all positive integers that, when written in binary, occur as substrings in binary k divide k.

1, 2, 3, 4, 6, 8, 10, 12, 16, 20, 24, 32, 36, 40, 48, 64, 72, 80, 96, 128, 136, 144, 160, 192, 256, 272, 288, 320, 384, 512, 528, 544, 576, 640, 768, 1024, 1056, 1088, 1152, 1280, 1536, 2048, 2080, 2112, 2176, 2304, 2560, 3072, 4096, 4160, 4224, 4352, 4608

Offset: 1

Author

Leroy Quet, Apr 24 2010

Keywords

• base
• nonn

Comments

This is the complement of sequence A175382.

This sequence is infinite (because it contains all powers of 2).

The union of powers of 2 and numbers of the form 2^n + 2^k, where floor(n/2) <= k < n. - Ivan Neretin, Nov 24 2016

Examples

			20 in binary is 10100. The positive integers that, when written in binary, occur as substrings in 10100 are: 1 (1 in binary), 2 (10 in binary), 4 (100 in binary), 5 (101 in binary), 10 (1010 in binary), and 20 (10100 in binary.) Since 1, 2, 4, 5, 10, and 20 each are a divisor of 20, then 20 is in this sequence.
		

Links

• Ivan Neretin, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A175382.

Programs

Mathematica

mx = 12; Union[2^Range[0, mx], Flatten@Table[2^n + 2^k, {n, 0, mx}, {k, Quotient[n, 2], n - 1}]] (* Ivan Neretin, Nov 24 2016 *)

Python

def ok(n):
    s = bin(n)[2:]
    subs = (s[i:j] for i in range(len(s)) for j in range(i+1, len(s)+1) if s[i]!='0')
    return n and all(n%v == 0 for ss in subs if (v:=int(ss, 2)) > 0)
print([k for k in range(5000) if ok(k)]) # Michael S. Branicky, May 09 2025

Extensions

Spelling corrected by Jason G. Wurtzel, Sep 04 2010

a(11)-a(53) from Lars Blomberg, May 05 2011