|
%I #14 Feb 22 2015 18:06:16
%S 1,2,3,4,5,5,7,8,9,9,11,10,13,13,14,16,17,15,19,18,20,21,23,20,25,25,
%T 27,26,29,22,31,32,32,33,34,30,37,37,38,36,41,32,43,42,42,45,47,40,49,
%U 45,50,50,53,45,54,52,56,57,59,44,61,61,60,64,64,52,67,66,68,58,71,60,73
%N a(n) = number of positive integers <= n which don't have 2 or more distinct prime divisors in common with n.
%C Equivalently, a(n) is the number of integers m, 1<=m<=n such that gcd(m,n) is 1 or a prime or a prime power, i.e. gcd(m,n)=p^k for some prime p and some k>=0. Cf. A117494. - _Geoffrey Critzer_, Feb 22 2015
%H Alois P. Heinz, <a href="/A131233/b131233.txt">Table of n, a(n) for n = 1..10000</a>
%F Dirichlet g.f.: A(s)*zeta(s-1)/zeta(s) where A(s)= Sum_{n>=1} A010055(n)/n^s - _Geoffrey Critzer_, Feb 22 2015
%e The distinct primes which divide 20 are 2 and 5. So a(20) is the number of positive integers <= 20 which are not divisible by at least 2 distinct primes dividing 20; i.e. are not divisible by both 2 and 5. Among the first 20 positive integers only 10 and 20 are divisible by both 2 and 5. There are 18 other positive integers <= 20, so a(20)=18.
%p with(numtheory):
%p a:= n-> add(`if`(nops(factorset(igcd(n,k)))<2, 1, 0), k=1..n):
%p seq(a(n), n=1..100); # _Alois P. Heinz_, Feb 22 2015
%t nn = 73; f[list_, i_] := list[[i]]; a =Table[If[Length[FactorInteger[n]] == 1, 1, 0], {n, 1, nn}]; b =Table[EulerPhi[n], {n, 1, nn}]; Table[
%t DirichletConvolve[f[a, n], f[b, n], n, m], {m, 1, nn}] (* _Geoffrey Critzer_, Feb 22 2015 *)
%Y Cf. A131232.
%K nonn
%O 1,2
%A _Leroy Quet_, Jun 20 2007
%E More terms from _Joshua Zucker_, Jul 18 2007
|
