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 A056172 Number of non-unitary 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,6 COMMENTS A non-unitary 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 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 EXAMPLE 10!=2*2*2*2*2*2*2*2*3*3*3*3*5*5*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. 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

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Last modified November 30 02:32 EST 2022. Contains 358431 sequences. (Running on oeis4.)