|
| |
|
|
A056170
|
|
Number of non-unitary prime divisors of n.
|
|
41
|
|
|
|
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
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
|
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
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)
|
|
|
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 *)
|
|
|
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, A048105, A056169, A124010, A162641, A212177, A295659, A295666.
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
|
| |
|
|
