A353068 Irregular triangle read by rows: T(n,k) = n - multiplicity of prime(k) as a divisor of n!.

1, 2, 2, 1, 3, 2, 4, 4, 2, 4, 5, 3, 5, 6, 6, 1, 6, 7, 7, 2, 5, 8, 8, 2, 6, 8, 9, 3, 7, 9, 10, 10, 2, 7, 10, 11, 11, 3, 8, 11, 12, 12, 12, 3, 9, 12, 12, 13, 13, 4, 9, 12, 13, 14, 14, 1, 10, 13, 14, 15, 15, 2, 11, 14, 15, 16, 16, 16, 2, 10, 15, 16, 17, 17, 17, 3, 11, 16, 17, 18, 18, 18, 18

Offset: 2

Author

Michel Marcus, Apr 21 2022

Keywords

• nonn
• tabf

Comments

This is the "s" in the American Mathematical Monthly problem.

Examples

			First few rows are:
  1;
  2, 2;
  1, 3;
  2, 4, 4;
  2, 4, 5;
  3, 5, 6, 6;
  1, 6, 7, 7;
  2, 5, 8, 8;
  2, 6, 8, 9;
  3, 7, 9, 10, 10;
  2, 7, 10, 11, 11;
  ...
		

Links

• Michel Marcus, Table of n, a(n) for n = 2..10388 (Rows n=2..300).
• R. D. Carmichael, Diophantine Analysis: 141, The American Mathematical Monthly, Vol. 14, No. 5 (May, 1907), p. 107.

Crossrefs

Cf. A115627, A000120 (column 1), A089792 (column 2).

Programs

Mathematica

T[n_, k_] := n - IntegerExponent[n!, Prime[k]]; Table[T[n, k], {n, 2, 19}, {k, 1, PrimePi[n]}] // Flatten (* Amiram Eldar, Apr 21 2022 *)

PARI

row(n) = vector(primepi(n), k, n-valuation(n!, prime(k)));

Formula

T(n,k) = n - A115627(n, k).