A182910 Number of unitary prime divisors of the swinging factorial (A056040) n$ = n! / floor(n/2)!^2.

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

Offset: 0

Author

Peter Luschny, Mar 14 2011

Keywords

• nonn

Comments

A prime divisor of n is unitary iff its exponent is 1 in the prime power factorization of n. A unitary prime divisor of the swinging factorial n$ can be smaller than n/2. For n >= 30 the swinging factorial has more unitary prime divisors than the factorial and it never has fewer unitary prime divisors. Thus a(n) >= PrimePi(n) - PrimePi(n/2).

Examples

			16$ = 2*3*3*5*11*13. So 16$ has one non-unitary prime divisor and a(16) = 4.
		

Crossrefs

Cf. A056171.

Programs

Maple

UnitaryPrimeDivisor := proc(f,n) local k, F; F := f(n):
add(`if`(igcd(iquo(F,k),k)=1,1,0),k=numtheory[factorset](F)) end;
A056040 := n -> n!/iquo(n,2)!^2;
A182910 := n -> UnitaryPrimeDivisor(A056040,n);
seq(A182910(i), i=1..LEN);

Mathematica

Table[Function[m, If[m == 1, 0, Count[FactorInteger[m][[All, -1]], 1]]][n!/Floor[n/2]!^2], {n, 0, 67}] (* Michael De Vlieger, Aug 02 2017 *)

Python

from sympy import factorint, factorial
def a056169(n): return 0 if n==1 else sum(1 for i in factorint(n).values() if i==1)
def a056040(n): return factorial(n)//factorial(n//2)**2
def a(n): return a056169(a056040(n))
print([a(n) for n in range(68)]) # Indranil Ghosh, Aug 02 2017