A056170 Number of non-unitary prime divisors of n.

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

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Index entries for sequences computed from exponents in factorization of n.

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

Crossrefs

Cf. A000188, A001221, A003557, A013940, A034444, A046660, A048105, A056169, A085548, A124010, A162641, A212177, A275812, A295659, A295666.

Cf. A057427(a(n)) = 1 - A008966(n).

Sequence in context: A101436 A366247 A369165 * A248395 A059483 A067618

Adjacent sequences: A056167 A056168 A056169 * A056171 A056172 A056173

Keyword

nice,nonn

Author

Labos Elemer, Jul 27 2000

Extensions

Minor edits by Franklin T. Adams-Watters, Mar 23 2011

Status

approved