A378912 Irregular triangle read by rows: row n lists all positive m such that sigma(m) divides n, where sigma is the sum-of-divisors function (A000203).

1, 1, 1, 2, 1, 3, 1, 1, 2, 5, 1, 4, 1, 3, 7, 1, 2, 1, 1, 1, 2, 3, 5, 6, 11, 1, 9, 1, 4, 13, 1, 2, 8, 1, 3, 7, 1, 1, 2, 5, 10, 17, 1, 1, 3, 19, 1, 2, 4, 1, 1, 1, 2, 3, 5, 6, 7, 11, 14, 15, 23, 1, 1, 9, 1, 2, 1, 3, 4, 12, 13, 1, 1, 2, 5, 8, 29, 1, 16, 25, 1, 3, 7, 21, 31

Offset: 1

Author

Paolo Xausa, Dec 10 2024

Keywords

• nonn
• tabf

Examples

			Triangle begins:
  n\k|  1   2   3   4   5   6 ...
  -------------------------------
   1 |  1;
   2 |  1;
   3 |  1,  2;
   4 |  1,  3;
   5 |  1;
   6 |  1,  2,  5;
   7 |  1,  4;
   8 |  1,  3,  7;
   9 |  1,  2;
  10 |  1;
  11 |  1;
  12 |  1,  2,  3,  5,  6, 11;
  13 |  1,  9;
  14 |  1,  4, 13;
  15 |  1,  2,  8;
  16 |  1,  3,  7;
  17 |  1;
  18 |  1,  2,  5, 10, 17;
  19 |  1;
  20 |  1,  3, 19;
  ...
		

Links

• Paolo Xausa, Table of n, a(n) for n = 1..10634 (rows 1..2000 of triangle, flattened).
• Index entries for sequences related to sigma(n)

Crossrefs

Cf. A074754 (row lengths), A319068 (right border), A378913 (row sums).

Cf. A000203.

Programs

Mathematica

With[{nmax = 50}, Table[PositionIndex[Divisible[n, #[[;; n]]]][True], {n, nmax}] & [DivisorSigma[1, Range[nmax]]]]

PARI

row(n) = select(x->(!(n % sigma(x))), [1..n]); \\ Michel Marcus, Dec 11 2024

Formula

T(n,k) <= n (see A319068).