A355633 -id:A355633 - cp's OEIS frontend

A355620 a(n) is the sum of the divisors of n whose decimal expansions appear as substrings in the decimal expansion of n.

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 15, 14, 15, 21, 17, 18, 19, 20, 22, 22, 24, 23, 30, 30, 28, 27, 30, 29, 33, 32, 34, 36, 34, 40, 45, 37, 38, 42, 44, 42, 44, 43, 48, 50, 46, 47, 60, 49, 55, 52, 54, 53, 54, 60, 56, 57, 58, 59, 66, 62, 64, 66, 68, 70, 72, 67

Offset: 1

Rémy Sigrist, Jul 10 2022

			For n = 110:
- the divisors of 110 are: 1, 2, 5, 10, 11, 22, 55, 110,
- 1, 10, 11 and 110 appear as substrings in 110,
- so a(110) = 1 + 10 + 11 + 110 = 132.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Scatterplot of the first 100000 terms (red pixels indicate when n is a multiple of 10)
Index entries for sequences related to decimal expansion of n
Index entries for sequences related to divisors

Cf. A000203, A002275, A121041, A121042, A239058, A355633 (binary analog).

Table[DivisorSum[n, # &, StringContainsQ[IntegerString[n], IntegerString[#]] &], {n, 100}] (* Paolo Xausa, Jul 23 2024 *)

a(n, base=10) = { 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 = str(n)
    return sum(d for d in divisors(n, generator=True) if str(d) in s)
print([a(n) for n in range(1, 68)]) # Michael S. Branicky, Jul 10 2022

a(n) >= n.
a(n) <= A000203(n) with equality iff n belongs to A239058.
a(10^n) = A002275(n+1) for any n >= 0.

A355634 Irregular triangle T(n, k), n > 0, k = 1..A093640(n), read by rows; the n-th row contains in ascending order the divisors of n whose binary expansions appear as substrings in the binary expansion of n.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 2, 4, 1, 5, 1, 2, 3, 6, 1, 7, 1, 2, 4, 8, 1, 9, 1, 2, 5, 10, 1, 11, 1, 2, 3, 4, 6, 12, 1, 13, 1, 2, 7, 14, 1, 3, 15, 1, 2, 4, 8, 16, 1, 17, 1, 2, 9, 18, 1, 19, 1, 2, 4, 5, 10, 20, 1, 21, 1, 2, 11, 22, 1, 23, 1, 2, 3, 4, 6, 8, 12, 24, 1, 25

Offset: 1

Rémy Sigrist, Jul 11 2022

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

Paolo Xausa, Table of n, a(n) for n = 1..10019 (rows 1..2000 of the triangle, flattened).
Index entries for sequences related to binary expansion of n
Index entries for sequences related to divisors

Cf. A027750, A093640 (row lengths), A355632 (decimal analog), A355633 (row sums).

Table[Select[Divisors[n], StringContainsQ[IntegerString[n, 2], IntegerString[#, 2]] &], {n, 50}] (* Paolo Xausa, Jul 23 2024 *)

row(n, base=2) = { 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 = bin(n)[2:]
    return sorted(d for d in divisors(n, generator=True) if bin(d)[2:] in s)
def table(r): return [i for n in range(1, r+1) for i in row(n)]
print(table(25)) # Michael S. Branicky, Jul 11 2022

T(n, 1) = 1.
T(n, A093640(n)) = n.
Sum_{k = 1..A093640(n)} T(n, k) = A355633(n).

cp's OEIS Frontend

A355620 a(n) is the sum of the divisors of n whose decimal expansions appear as substrings in the decimal expansion of n.

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