A090623 Triangle of T(n,k) = [n/k] + [n/k^2] + [n/k^3] + [n/k^4] + ... for n, k > 1.

1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 4, 2, 1, 1, 1, 4, 2, 1, 1, 1, 1, 7, 2, 2, 1, 1, 1, 1, 7, 4, 2, 1, 1, 1, 1, 1, 8, 4, 2, 1, 1, 1, 1, 1, 1, 8, 4, 2, 1, 1, 1, 1, 1, 1, 1, 10, 5, 2, 1, 1, 1, 1, 1, 1, 1, 1, 10, 5, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 5, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 6, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset

2,4

Links

Table of n, a(n) for n=2..104.

Wenguang Zhai, On the prime power factorization of n!, Journal of Number Theory, Volume 129, Issue 8, August 2009, Pages 1820-1836.

Formula

For p prime, T(n, p) = A090622(n, p) is the number of times that p is a factor of n!.

Example

Rows start:
1;
1,1;
3,1,1;
3,1,1,1;
4,2,1,1,1;
4,2,1,1,1,1;
7,2,2,1,1,1,1;
7,4,2,1,1,1,1,1;
8,4,2,1,1,1,1,1,1;
etc.

Prog

(PARI) t(n, k) = {my(s = 0, j = 1); while(p=n\k^j, s += p; j++); s; } \\ Michel Marcus, Feb 02 2016

Crossrefs

Rows include A011371, A054861, A054893, A027868, A054895, A054896, A054897, A054898, A054899, A064458, A064459, A090620, A054900.

Sequence in context: A058965 A226306 A124921 * A098094 A367824 A087283

Adjacent sequences: A090620 A090621 A090622 * A090624 A090625 A090626

Keyword

nonn,tabl

Author

Henry Bottomley, Dec 06 2003

Status

approved