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<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) is also 1 + the number of divisors of n with omega(d)=1, where omega is A001221. - Enrique Pérez Herrero, Nov 05 2009
Length of n-th row of triangle A210208. - Reinhard Zumkeller, Mar 18 2012
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
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) = if n=1 then 1 else a(A032742(n)) + 1. - Reinhard Zumkeller, Sep 24 2009
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
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))
(PARI) A073093(n)=bigomega(n)+1 \\ M. F. Hasler, Dec 08 2010
(MuPAD) numlib::Omega (2*n)$ n=1..105 // Zerinvary Lajos, May 13 2008
(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
KEYWORD
nonn,easy
AUTHOR
Reinhard Zumkeller, Aug 24 2002
STATUS
approved