A056172 Number of non-unitary prime divisors of n!.

0, 0, 0, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 14, 14, 14

Offset: 1

Author

Labos Elemer, Jul 27 2000

Keywords

• nonn
• easy

Comments

A non-unitary prime divisor for n! cannot exceed n/2.

Examples

			10! = 2^8 * 3^4 * 5^2 * 7. The non-unitary prime divisors are 2, 3, and 5 because their exponents exceed 1, so a(10) = 3.  The only unitary prime divisor of 10! is 7.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000720, A001221, A034444, A048105, A048656, A048657, A056170, A056171.

Programs

Maple

with(numtheory); A056172:=n->pi(floor(n/2)); seq(A056172(k),k=1..100); # Wesley Ivan Hurt, Sep 30 2013

Mathematica

Table[PrimePi[Floor[n/2]], {n,100}] (* Wesley Ivan Hurt, Sep 30 2013 *)

Formula

a(n) = pi(n/2).

A prime divisor of x is non-unitary iff its exponent is at least 2 in the prime power factorization of x. In general, GCD(p, x/p) = 1 or p. Cases are counted when GCD(p, n/p) > 1.

a(n) = A000720(n) - A056171(n). - Robert G. Wilson v, Apr 09 2017

a(n) = A056170(n!). - Amiram Eldar, Jul 24 2024

Extensions

Example corrected by Jon E. Schoenfield, Sep 30 2013