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A056170 Number of non-unitary prime divisors of n. 33
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,36

COMMENTS

Number of prime squares 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

LINKS

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

FORMULA

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.)

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

Conjecture: a(n) = log_2(A000005(A071773(n))). - Velin Yanev, Aug 20 2017

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-Fran├žois Alcover, Mar 23 2011 *)

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. A001221, A013940, A034444, A048105, A124010, A212177.

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

Sequence in context: A093956 A160383 A101436 * 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

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Last modified September 24 17:02 EDT 2017. Contains 292432 sequences.