A355634 -id:A355634 - cp's OEIS frontend

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

Rémy Sigrist, Jul 11 2022

			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.

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

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

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

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) }

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

a(n) <= A000203(n).

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

A355632 Irregular triangle T(n, k), n > 0, k = 1..A121041(n), read by rows; the n-th row contains in ascending order the divisors of n whose decimal expansions appear as substrings in the decimal expansion of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 10, 1, 11, 1, 2, 12, 1, 13, 1, 14, 1, 5, 15, 1, 16, 1, 17, 1, 18, 1, 19, 2, 20, 1, 21, 2, 22, 23, 2, 4, 24, 5, 25, 2, 26, 27, 2, 28, 29, 3, 30, 1, 31, 2, 32, 3, 33, 34, 5, 35, 3, 6, 36, 37, 38, 3, 39, 4, 40, 1, 41, 2, 42, 43, 4, 44

Offset: 1

Rémy Sigrist, Jul 11 2022

			Triangle T(n, k) begins:
     1: [1]
     2: [2]
     3: [3]
     4: [4]
     5: [5]
     6: [6]
     7: [7]
     8: [8]
     9: [9]
    10: [1, 10]
    11: [1, 11]
    12: [1, 2, 12]
    13: [1, 13]
    14: [1, 14]
    15: [1, 5, 15]
    16: [1, 16]

Paolo Xausa, Table of n, a(n) for n = 1..12288 (rows 1..5000 of the triangle, flattened).
Index entries for sequences related to decimal expansion of n
Index entries for sequences related to divisors

Cf. A027750, A121041 (row lengths), A121042, A355620 (row sums), A355634 (binary analog).

Table[Select[Divisors[n], StringContainsQ[IntegerString[n], IntegerString[#]] &], {n, 50}] (* Paolo Xausa, Jul 23 2024 *)

row(n, base=10) = { my (d=digits(n, base), s=setbinop((i,j) -> fromdigits(d[i..j], base), [1..#d]), v=0); select(v -> v && n%v==0, s) }

from sympy import divisors
def row(n):
    s = str(n)
    return sorted(d for d in divisors(n, generator=True) if str(d) in s)
def table(r): return [i for n in range(1, r+1) for i in row(n)]
print(table(44)) # Michael S. Branicky, Jul 11 2022

T(n, 1) = A121042(n).
T(n, A121041(n)) = n.
Sum_{k = 1..A121041(n)} T(n, k) = A355620(n).

cp's OEIS Frontend

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

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