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A073093
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Number of prime power divisors of n.
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42
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1, 2, 2, 3, 2, 3, 2, 4, 3, 3, 2, 4, 2, 3, 3, 5, 2, 4, 2, 4, 3, 3, 2, 5, 3, 3, 4, 4, 2, 4, 2, 6, 3, 3, 3, 5, 2, 3, 3, 5, 2, 4, 2, 4, 4, 3, 2, 6, 3, 4, 3, 4, 2, 5, 3, 5, 3, 3, 2, 5, 2, 3, 4, 7, 3, 4, 2, 4, 3, 4, 2, 6, 2, 3, 4, 4, 3, 4, 2, 6, 5, 3, 2, 5, 3, 3, 3, 5, 2, 5, 3, 4, 3, 3, 3, 7, 2, 4, 4, 5, 2, 4, 2, 5, 4
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OFFSET
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1,2
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COMMENTS
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Also, number of prime divisors of 2n (counted with multiplicity).
a(n) is also the number of k<n such that the resultant of the k-th cyclotomic polynomial and the n-th cyclotomic polynomial is not 1. It is well known that if (k,n)=1, res(polcyclo(n),polcyclo(k))=1. -Benoit Cloitre,Oct 13 2002
a(n) depends only on the prime signature of n with a(A025487(n)) = 1, 2, 3, 3, 4, 4, 5, 5, 4, 6, 5, 6, 5, 7, 6, 7,.. =A036041(n)+1; (n>=1). -R. J. Mathar,May 28 2017
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LINKS
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FORMULA
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If n = Product (p_j^k_j), a(n) = 1 + Sum (k_j).
a(n) = max { a(d); d<n and d|n } + 1, if n > 1. -David W. Wilson,Dec 08 2010
G.f.: x/(1 - x) + Sum_{k>=2} floor(1/ Omega (k))*x^k/(1 - x^k), where Omega (k) is the number of distinct prime factors (A001221). -Ilya Gutkovskiy,Jan 04 2017
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MAPLE
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seq(numtheory:-big Omega (n)+1, n=1..1000); #Robert Israel,Sep 06 2015
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MATHEMATICA
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f[n_]:= Plus @@ Flatten[ Table[1, {#[[2]]}] & /@ FactorInteger[n]]; Table[ f[2n], {n, 105}] (*Robert G. Wilson v,Dec 23 2004 *)
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PROG
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(PARI) a(n)=sum(k=1, n, if(1-polresultant(polcyclo(n), polcyclo(k)), 1, 0))
(Haskell)
(Magma) [n eq 1 select 1 else &+[p[2]: p in Factorization(n)]+1: n in [1..100]]; //Vincenzo Librandi,Jan 06 2017
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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