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

1,2

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

Formula

a(n) <= A000203(n).

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

Example

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.

Mathematica

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

Prog

(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

Crossrefs

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

Sequence in context: A344575 A254981 A116607 * A107749 A353783 A367466

Adjacent sequences: A355630 A355631 A355632 * A355634 A355635 A355636

Keyword

nonn,base

Author

Rémy Sigrist, Jul 11 2022

Status

approved