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 A073093 Number of prime power divisors of n. 13
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Also, number of prime divisors of 2n (counted with multiplicity). A001221(n) < a(n) <= A000005(n) for all n; a(n)=A001221(n)+1 iff n is squarefree (A005117); a(n)=A000005(n) iff n is a prime power (A000961). a(n) is also the number of k=1). - R. J. Mathar, May 28 2017 LINKS Reinhard Zumkeller, Table of n, a(n) for n = 1..10000 T. M. Apostol, Resultants of Cyclotomic Polynomials, Proc. Amer. Math. Soc. 24, 457-462, 1970. T. M. Apostol, The Resultant of the Cyclotomic Polynomials Fm(ax) and Fn(bx), Math. Comput. 29, 1-6, 1975. Eric Weisstein's World of Mathematics, Cyclotomic Polynomial FORMULA If n = Product (p_j^k_j), a(n) = 1 + Sum (k_j). a(n) = bigomega(n)+1 = A001222(n)+1 = A001222(2*n). a(n) = if n=1 then 1 else a(A032742(n)) + 1. - Reinhard Zumkeller, Sep 24 2009 a(n) = max { a(d) ; d 1. - David W. Wilson, Dec 08 2010 a(n) = Sum_{k = 1 .. A001221(n)} A010055(A027750(n,k)). - Reinhard Zumkeller, Mar 18 2012 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 MAPLE seq(numtheory:-bigomega(n)+1, n=1..1000); # Robert Israel, Sep 06 2015 MATHEMATICA f[n_] := Plus @@ Flatten[ Table[1, {#[[2]]}] & /@ FactorInteger[n]]; Table[ f[2n], {n, 105}] (* Robert G. Wilson v, Dec 23 2004 *) A001221[n_] := (Length[ FactorInteger[n]]); SetAttributes[A001221, Listable]; A073093[n_]:=Length[Select[A001221[Divisors[n]], # == 1 &]]; (* Enrique Pérez Herrero, Nov 05 2009 *) PROG (PARI) a(n)=sum(k=1, n, if(1-polresultant(polcyclo(n), polcyclo(k)), 1, 0)) (Mupad) numlib::Omega (2*n)\$ n=1..105 // Zerinvary Lajos, May 13 2008 (PARI) A073093(n)=bigomega(n)+1   \\ M. F. Hasler, Dec 08 2010 (Haskell) a073093 = length . a210208_row  -- Reinhard Zumkeller, Mar 18 2012 (MAGMA) [n eq 1 select 1 else &+[p[2]: p in Factorization(n)]+1: n in [1..100]]; // Vincenzo Librandi, Jan 06 2017 CROSSREFS Cf. A000961, A023888, A054372. Bisection of A001222. Sequence in context: A052180 A065151 A175193 * A222084 A088873 A085082 Adjacent sequences:  A073090 A073091 A073092 * A073094 A073095 A073096 KEYWORD nonn,easy AUTHOR Reinhard Zumkeller, Aug 24 2002 STATUS approved

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