A244342 a(n) = phi(n)*h(n) where phi() is the Euler totient function, A000010, and h() is A092089.

1, 2, 6, 8, 12, 12, 18, 32, 30, 24, 30, 48, 36, 36, 72, 96, 48, 60, 54, 96, 108, 60, 66, 192, 100, 72, 126, 144, 84, 144, 90, 256, 180, 96, 216, 240, 108, 108, 216, 384, 120, 216, 126, 240, 360, 132, 138, 576, 210, 200, 288, 288, 156, 252, 360, 576, 324, 168

Offset: 1

Author

Michel Marcus, Jun 26 2014

Keywords

• nonn
• mult

Comments

a(n) = Sum_{k=1..n} gcd(k^2-1, n) for those k that are coprime to n (see proof in link).

Multiplicative because both A000010 and A092089 are. - Andrew Howroyd, Jul 26 2018

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• László Tóth, Menon's identity and arithmetical sums representing functions of several variables, Rend. Sem. Mat. Univ. Politec. Torino, 69 (2011), 97-110 (see (36) in Corollary 15, p. 108); also on arXiv, arXiv:1103.5861 [math.NT], 2011.

Crossrefs

Cf. A000010, A062355, A092089.

Programs

Maple

A244342:= proc(n) add(`if`(igcd(k,n)=1,igcd(k^2-1,n),0),k=1..n) end proc;
seq(A244342(i),i=1..1000); # Robert Israel, Jul 06 2014

Mathematica

h[n_] := Product[{p, e} = pe; Which[OddQ[p], 2 e + 1, p == 2 && e == 1, 2, True, 4 (e - 1)], {pe, FactorInteger[n]}]; h[1] = 1;
a[n_] := EulerPhi[n] h[n];
Array[a, 100] (* Jean-François Alcover, Apr 08 2020 *)

PARI

a(n) = sum(j=1, n, gcd(j^2-1,n)*(gcd(j,n)==1));