1,2

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

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

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

From Amiram Eldar, Jun 21 2025: (Start)

a(n) = A116512(n) + A000010(n).

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = (1 + Sum_{p prime} (1/(p^2-1))) / zeta(2) = (1 + A154945)/A013661 = 0.94331640941093700227... . (End)

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.

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

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 *)

a[n_] := Module[{p = FactorInteger[n][[;; , 1]]}, n * Times @@ (1-1/p) * (1 + Total[1/(p-1)])]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Jun 21 2025 *)

(PARI) a(n) = {my(p = factor(n)[, 1]); n * vecprod(apply(x -> 1-1/x, p)) * (1 + vecsum(apply(x -> 1/(x-1), p))); } \\ Amiram Eldar, Jun 21 2025

Cf. A000010, A010055, A013661, A131232, A116512, A117494, A154945.

Sequence in context: A162683 A073137 A345965 * A136623 A031218 A385199

Adjacent sequences: A131230 A131231 A131232 * A131234 A131235 A131236

Leroy Quet, Jun 20 2007

More terms from Joshua Zucker, Jul 18 2007

approved