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

2,2

Comments

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

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.

Formula

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

Example

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;
  ...

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 *)

Prog

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

Crossrefs

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

Sequence in context: A375300 A291712 A074945 * A367292 A323479 A276427

Adjacent sequences: A353065 A353066 A353067 * A353069 A353070 A353071

Keyword

nonn,tabf

Author

Michel Marcus, Apr 21 2022

Status

approved