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

Author

Rémy Sigrist, Jul 10 2022

Keywords

• nonn
• base

Examples

			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.
		

Links

• 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

Crossrefs

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

Programs

Mathematica

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

PARI

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) }

Python

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

Formula

a(n) >= n.

a(n) <= A000203(n) with equality iff n belongs to A239058.

a(10^n) = A002275(n+1) for any n >= 0.