

A131233


a(n) = number of positive integers <= n which don't have 2 or more distinct prime divisors in common with n.


2



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, 27, 26, 29, 22, 31, 32, 32, 33, 34, 30, 37, 37, 38, 36, 41, 32, 43, 42, 42, 45, 47, 40, 49, 45, 50, 50, 53, 45, 54, 52, 56, 57, 59, 44, 61, 61, 60, 64, 64, 52, 67, 66, 68, 58, 71, 60, 73
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OFFSET

1,2


COMMENTS

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


LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..10000


FORMULA

Dirichlet g.f.: A(s)*zeta(s1)/zeta(s) where A(s)= Sum_{n>=1} A010055(n)/n^s  Geoffrey Critzer, Feb 22 2015


EXAMPLE

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.


MAPLE

with(numtheory):
a:= n> add(`if`(nops(factorset(igcd(n, k)))<2, 1, 0), k=1..n):
seq(a(n), n=1..100); # Alois P. Heinz, Feb 22 2015


MATHEMATICA

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[
DirichletConvolve[f[a, n], f[b, n], n, m], {m, 1, nn}] (* Geoffrey Critzer, Feb 22 2015 *)


CROSSREFS

Cf. A131232.
Sequence in context: A225090 A162683 A073137 * A136623 A031218 A267508
Adjacent sequences: A131230 A131231 A131232 * A131234 A131235 A131236


KEYWORD

nonn


AUTHOR

Leroy Quet, Jun 20 2007


EXTENSIONS

More terms from Joshua Zucker, Jul 18 2007


STATUS

approved



