OFFSET
0,4
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).
EXAMPLE
16$ = 2*3*3*5*11*13. So 16$ has one non-unitary prime divisor and a(16) = 4.
MAPLE
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Peter Luschny, Mar 14 2011
STATUS
approved