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 A056170 Number of non-unitary prime divisors of n. 58
 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 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 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 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; [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, A085548, A124010, A162641, A212177, A295659, A295666. Cf. A057427(a(n)) = 1 - A008966(n). Sequence in context: A330023 A328891 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 November 30 12:40 EST 2022. Contains 358441 sequences. (Running on oeis4.)