A356555 - OEIS

%I #10 Aug 15 2022 05:22:59

%S 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,

%T 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,

%U 11,13,15,16,2,3,4,5,7,8,9,13,15,16,17,17,18

%N 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.

%C 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).

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

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

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

%e Triangle T(n, k) begins:

%e     n    n-th row

%e     --   --------

%e      1   [2]

%e      2   [2, 3]

%e      3   [3, 4]

%e      4   [2, 3, 4, 5]

%e      5   [5, 6]

%e      6   [2, 3, 4, 5, 6, 7]

%e      7   [7, 8]

%e      8   [2, 3, 4, 5, 7, 8, 9]

%e      9   [3, 4, 7, 9, 10]

%e     10   [2, 3, 5, 6, 9, 10, 11]

%e     11   [11, 12]

%e     12   [2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13]

%e     13   [13, 14]

%e     14   [7, 8, 13, 14, 15]

%e     15   [3, 5, 6, 7, 11, 13, 15, 16]

%e     16   [2, 3, 4, 5, 7, 8, 9, 13, 15, 16, 17]

%e     17   [17, 18]

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

%o (Python)

%o from sympy.ntheory import digits

%o def row(n): return [b for b in range(2, n+2) if n%sum(digits(n, b)[1:])==0]

%o print([an for n in range(1, 18) for an in row(n)]) # _Michael S. Branicky_, Aug 12 2022

%Y Cf. A080221, A356552.

%K nonn,base,tabf

%O 1,1

%A _Rémy Sigrist_, Aug 12 2022