A355633 a(n) is the sum of the divisors of n whose binary expansions appear as substrings in the binary expansion of n.

1, 3, 4, 7, 6, 12, 8, 15, 10, 18, 12, 28, 14, 24, 19, 31, 18, 30, 20, 42, 22, 36, 24, 60, 26, 42, 31, 56, 30, 57, 32, 63, 34, 54, 36, 70, 38, 60, 43, 90, 42, 66, 44, 84, 54, 72, 48, 124, 50, 78, 55, 98, 54, 93, 72, 120, 61, 90, 60, 133, 62, 96, 74, 127, 66

Offset: 1

Author

Rémy Sigrist, Jul 11 2022

Keywords

• nonn
• base

Examples

			For n = 84:
- the binary expansion of 84 is "1010100",
- we have the following divisors:
  d   bin(d)   in bin(84)?
  --  -------  -----------
   1        1  Yes
   2       10  Yes
   3       11  No
   4      100  Yes
   6      110  No
   7      111  No
  12     1100  No
  14     1110  No
  21    10101  Yes
  28    11100  No
  42   101010  Yes
  84  1010100  Yes
- so a(84) = 1 + 2 + 4 + 21 + 42 + 84 = 154.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000
• Rémy Sigrist, Colored scatterplot of the first 100000 terms (the color is function of the 2-adic valuation of n)
• Index entries for sequences related to binary expansion of n
• Index entries for sequences related to divisors

Crossrefs

Cf. A000203, A027750, A093640, A355620 (decimal analog), A355634.

Programs

Mathematica

a[n_] := DivisorSum[n, # &, StringContainsQ @@ IntegerString[{n, #}, 2] &]; Array[a, 100] (* Amiram Eldar, Jul 16 2022 *)

PARI

a(n, base=2) = { my (d=digits(n, base), s=setbinop((i, j) -> fromdigits(d[i..j], base), [1..#d]), v=0); for (k=1, #s, if (s[k] && n%s[k]==0, v+=s[k])); return (v) }

Python

from sympy import divisors
def a(n):
    s = bin(n)[2:]
    return sum(d for d in divisors(n, generator=True) if bin(d)[2:] in s)
print([a(n) for n in range(1, 66)]) # Michael S. Branicky, Jul 11 2022

Formula

a(n) <= A000203(n).

a(2^n) = 2^(n+1) - 1 for any n >= 0.