Search: id:A073093
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%I A073093
%S A073093 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,
%T A073093 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,
%U A073093 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
%N A073093 Number of prime power divisors of n.
%C A073093 Also, number of prime divisors of 2n (counted with multiplicity).
%C A073093 A001221(n) < a(n) <= A000005(n) for all n; a(n)=A001221(n)+1 iff n is 
               square-free (A005117); a(n)=A000005(n) iff n is a prime power (A000961).
%C A073093 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.
%C A073093 a(n) is also the number of divisors of n with omega(d)=1, where omega 
               is A001221 [From Barbarel Tres Mil (barbarel3000(AT)yahoo.es), Nov 
               05 2009]
%D A073093 Apostol, T. M. Resultants of Cyclotomic Polynomials. Proc. Amer. Math. 
               Soc. 24, 457-462, 1970.
%D A073093 Apostol, T. M. The Resultant of the Cyclotomic Polynomials ..., Math. 
               Comput. 29, 1-6, 1975.
%H A073093 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               CyclotomicPolynomial.html">Link to a section of The World of Mathematics.</
               a>
%F A073093 If n = Product (p_j^k_j), a(n) = 1 + Sum (k_j).
%F A073093 a(n) = if n=1 then 1 else a(A032742(n)) + 1. [From Reinhard Zumkeller 
               (reinhard.zumkeller(AT)gmail.com), Sep 24 2009]
%t A073093 f[n_] := Plus @@ Flatten[ Table[1, {#[[2]]}] & /@ FactorInteger[n]]; 
               Table[ f[2n], {n, 105}] (from Robert G. Wilson v Dec 23 2004)
%t A073093 A001221[n_] := (Length[ FactorInteger[n]]); SetAttributes[A001221, Listable]; 
               A073093[n_]:=Length[Select[A001221[Divisors[n]], # == 1 &]]; [From 
               Barbarel Tres Mil (barbarel3000(AT)yahoo.es), Nov 05 2009]
%o A073093 (PARI) a(n)=sum(k=1,n,if(1-polresultant(polcyclo(n),polcyclo(k)),1,0))
%o A073093 (Mupad) numlib::Omega (2*n)$ n=1..105 - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               May 13 2008
%Y A073093 Cf. A023888, A000961, A054372. Bisection of A001222.
%Y A073093 a(n) = bigomega(n)+1 = bigomega(2n), cf. A001222.
%Y A073093 Sequence in context: A087458 A052180 A065151 this_sequence A088873 A085082 
               A067554
%Y A073093 Adjacent sequences: A073090 A073091 A073092 this_sequence A073094 A073095 
               A073096
%K A073093 nonn,easy,new
%O A073093 1,2
%A A073093 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Aug 24 2002

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