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A056170
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Number of non-unitary prime divisors of n.
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73
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0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 2, 0, 0, 0, 1, 0
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OFFSET
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1,36
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COMMENTS
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A prime factor of n is unitary iff its exponent is 1 in the prime factorization of n. (Of course for any prime p, GCD(p, n/p) is either 1 or p. For a unitary prime factor it must be 1.)
Number of exponents larger than 1 in the prime factorization of n. - Antti Karttunen, Nov 28 2017
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LINKS
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FORMULA
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Additive with a(p^e) = 0 if e = 1, 1 otherwise.
G.f.: Sum_{k>=1} x^(prime(k)^2)/(1 - x^(prime(k)^2)). - Ilya Gutkovskiy, Jan 01 2017
For all n >= 1 it holds that:
(End)
Asymptotic mean: lim_{n->oo} (1/n) * Sum_{k=1..n} a(k) = Sum_{p prime} 1/p^2 = 0.452247... (A085548). - Amiram Eldar, Nov 01 2020
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MAPLE
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A056170 := n -> nops(select(t -> (t[2]>1), ifactors(n)[2]));
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MATHEMATICA
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a[n_] := Count[FactorInteger[n], {_, k_ /; k > 1}]; Table[a[n], {n, 105}] (* Jean-François Alcover, Mar 23 2011 *)
Table[Count[FactorInteger[n][[All, 2]], _?(#>1&)], {n, 110}] (* Harvey P. Dale, Jul 08 2019 *)
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PROG
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(Haskell)
a056170 = length . filter (> 1) . a124010_row
(Magma)
A056170:=func<n|#[pe:pe in Factorisation(n)|pe[2]ne 1]>;
(Python)
from sympy import factorint
def a(n):
f = factorint(n)
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CROSSREFS
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Cf. A000188, A001221, A003557, A013940, A034444, A046660, A048105, A056169, A085548, A124010, A162641, A212177, A275812, A295659, A295666.
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KEYWORD
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nice,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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