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.

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

Author

Rémy Sigrist, Jul 11 2022

Keywords

• nonn
• base
• tabf

Examples

			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]
		

Links

• 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

Crossrefs

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

Programs

Mathematica

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

PARI

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) }

Python

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

Formula

T(n, 1) = A121042(n).

T(n, A121041(n)) = n.

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