A356555 Irregular triangle T(n, k), n > 0, k = 1..A080221(n) read by rows; the n-th row contains, in ascending order, the bases b from 2..n+1 where the sum of digits of n divides n.

2, 2, 3, 3, 4, 2, 3, 4, 5, 5, 6, 2, 3, 4, 5, 6, 7, 7, 8, 2, 3, 4, 5, 7, 8, 9, 3, 4, 7, 9, 10, 2, 3, 5, 6, 9, 10, 11, 11, 12, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 13, 14, 7, 8, 13, 14, 15, 3, 5, 6, 7, 11, 13, 15, 16, 2, 3, 4, 5, 7, 8, 9, 13, 15, 16, 17, 17, 18

Offset: 1

Author

Rémy Sigrist, Aug 12 2022

Keywords

• nonn
• base
• tabf

Comments

A080221 provides row lengths (note that for n > 0, we consider the base n+1 but not the base 1, unlike A080221 that considers the base 1 but not the base n+1, however this does not matter as the sums of digits of n in base 1 and base n+1 are the same).

Examples

			Triangle T(n, k) begins:
    n    n-th row
    --   --------
     1   [2]
     2   [2, 3]
     3   [3, 4]
     4   [2, 3, 4, 5]
     5   [5, 6]
     6   [2, 3, 4, 5, 6, 7]
     7   [7, 8]
     8   [2, 3, 4, 5, 7, 8, 9]
     9   [3, 4, 7, 9, 10]
    10   [2, 3, 5, 6, 9, 10, 11]
    11   [11, 12]
    12   [2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13]
    13   [13, 14]
    14   [7, 8, 13, 14, 15]
    15   [3, 5, 6, 7, 11, 13, 15, 16]
    16   [2, 3, 4, 5, 7, 8, 9, 13, 15, 16, 17]
    17   [17, 18]
		

Crossrefs

Cf. A080221, A356552.

Programs

PARI

row(n) = select(b -> n % sumdigits(n,b)==0, [2..n+1])

Python

from sympy.ntheory import digits
def row(n): return [b for b in range(2, n+2) if n%sum(digits(n, b)[1:])==0]
print([an for n in range(1, 18) for an in row(n)]) # Michael S. Branicky, Aug 12 2022

Formula

T(n, 1) = A356552(n).

T(n, A080221(n)-1) = n for n > 1.

T(n, A080221(n)) = n+1.