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
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
LINKS
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) = omega(A000188(n)) = omega(A003557(n)) = omega(A057521(n)) = omega(A295666(n)), where omega = A001221.
For all n >= 1 it holds that:
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
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
CROSSREFS
KEYWORD
nice,nonn
AUTHOR
Labos Elemer, Jul 27 2000
EXTENSIONS
Minor edits by Franklin T. Adams-Watters, Mar 23 2011
STATUS
approved