|
Search: id:A073093
|
|
|
| A073093 |
|
Number of prime power divisors of n. |
|
+0
6
|
|
| 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; listen)
|
|
|
OFFSET
|
1,2
|
|
|
COMMENT
|
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 square-free (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 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]
|
|
REFERENCES
|
Apostol, T. M. Resultants of Cyclotomic Polynomials. Proc. Amer. Math. Soc. 24, 457-462, 1970.
Apostol, T. M. The Resultant of the Cyclotomic Polynomials ..., Math. Comput. 29, 1-6, 1975.
|
|
LINKS
|
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
|
|
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. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Sep 24 2009]
|
|
MATHEMATICA
|
f[n_] := Plus @@ Flatten[ Table[1, {#[[2]]}] & /@ FactorInteger[n]]; Table[ f[2n], {n, 105}] (from Robert G. Wilson v Dec 23 2004)
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]
|
|
PROGRAM
|
(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 (zerinvarylajos(AT)yahoo.com), May 13 2008
|
|
CROSSREFS
|
Cf. A023888, A000961, A054372. Bisection of A001222.
a(n) = bigomega(n)+1 = bigomega(2n), cf. A001222.
Sequence in context: A087458 A052180 A065151 this_sequence A088873 A085082 A067554
Adjacent sequences: A073090 A073091 A073092 this_sequence A073094 A073095 A073096
|
|
KEYWORD
|
nonn,easy
|
|
AUTHOR
|
Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Aug 24 2002
|
|
|
Search completed in 0.002 seconds
|
