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

1,6

Comments

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

Links

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

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

Example

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.

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 *)

Crossrefs

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

Sequence in context: A326032 A169990 A055679 * A285881 A091373 A197637

Adjacent sequences: A056169 A056170 A056171 * A056173 A056174 A056175

Keyword

nonn,easy

Author

Labos Elemer, Jul 27 2000

Extensions

Example corrected by Jon E. Schoenfield, Sep 30 2013

Status

approved