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 A182910 Number of unitary prime divisors of the swinging factorial (A056040) n\$ = n! / floor(n/2)!^2. 1
 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 (list; graph; refs; listen; history; text; internal format)
 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). LINKS EXAMPLE 16\$ = 2*3*3*5*11*13. So 16\$ has one non-unitary prime divisor and a(16) = 4. 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 *) PROG (Python) from sympy import factorint, factorial, floor 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(floor(n/2))**2 def a(n): return a056169(a056040(n)) print map(a, xrange(68)) # Indranil Ghosh, Aug 02 2017 CROSSREFS Cf. A056171. Sequence in context: A226743 A166269 A181648 * A055460 A067514 A115323 Adjacent sequences:  A182907 A182908 A182909 * A182911 A182912 A182913 KEYWORD nonn AUTHOR Peter Luschny, Mar 14 2011 STATUS approved

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Last modified October 18 17:13 EDT 2019. Contains 328186 sequences. (Running on oeis4.)