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A056170
Number of non-unitary prime divisors of n.
74
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
OFFSET
1,36
COMMENTS
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 squared primes dividing n. - Reinhard Zumkeller, May 18 2002
a(A005117(n)) = 0; a(A013929(n)) > 0; a(A190641(n)) = 1. - Reinhard Zumkeller, Dec 29 2012
First differences of A013940. - Jason Kimberley, Feb 01 2017
Number of exponents larger than 1 in the prime factorization of n. - Antti Karttunen, Nov 28 2017
FORMULA
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
a(n) = log_2(A000005(A071773(n))). - observed by Velin Yanev, Aug 20 2017, confirmed by Antti Karttunen, Nov 28 2017
From Antti Karttunen, Nov 28 2017: (Start)
a(n) = A001221(n) - A056169(n).
a(n) = omega(A000188(n)) = omega(A003557(n)) = omega(A057521(n)) = omega(A295666(n)), where omega = A001221.
For all n >= 1 it holds that:
a(A003557(n)) = A295659(n).
a(n) >= A162641(n).
(End)
Dirichlet g.f.: primezeta(2s)*zeta(s). - Benedict W. J. Irwin, Jul 11 2018
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
a(n) = A275812(n) - A046660(n). - Amiram Eldar, Jan 09 2024
MAPLE
A056170 := n -> nops(select(t -> (t[2]>1), ifactors(n)[2]));
seq(A056170(n), n=1..100); # Robert Israel, Jun 03 2014
MATHEMATICA
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 *)
PROG
(Haskell)
a056170 = length . filter (> 1) . a124010_row
-- Reinhard Zumkeller, Dec 29 2012
(PARI) a(n)=my(f=factor(n)[, 2]); sum(i=1, #f, f[i]>1) \\ Charles R Greathouse IV, May 18 2015
(Magma)
A056170:=func<n|#[pe:pe in Factorisation(n)|pe[2]ne 1]>;
[A056170(n):n in[1..105]];
// Jason Kimberley, Jan 22 2017
(Python)
from sympy import factorint
def a(n):
f = factorint(n)
return sum([1 for i in f if f[i]!=1]) # Indranil Ghosh, Apr 24 2017
KEYWORD
nice,nonn
AUTHOR
Labos Elemer, Jul 27 2000
EXTENSIONS
Minor edits by Franklin T. Adams-Watters, Mar 23 2011
STATUS
approved