

A056172


Number of nonunitary prime divisors of n!.


14



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
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OFFSET

1,6


COMMENTS

A nonunitary prime divisor for n! cannot exceed n/2. a(n) =PrimePi(n/2).


LINKS

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


FORMULA

A prime divisor of x is nonunitary 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


EXAMPLE

10!=2*2*2*2*2*2*2*2*3*3*3*3*5*5*7. The nonunitary 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. A001221, A034444, A000720, A048105, A048656, A048657.
Sequence in context: A326032 A169990 A055679 * A285881 A091373 A197637
Adjacent sequences: A056169 A056170 A056171 * A056173 A056174 A056175


KEYWORD

nonn


AUTHOR

Labos Elemer, Jul 27 2000


EXTENSIONS

Example corrected by Jon E. Schoenfield, Sep 30 2013


STATUS

approved



