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

1,2

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.

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 *)

Prog

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

Crossrefs

Cf. A000010, A062355, A092089.

Sequence in context: A130205 A054067 A250190 * A064212 A056906 A257056

Adjacent sequences: A244339 A244340 A244341 * A244343 A244344 A244345

Keyword

nonn,mult

Author

Michel Marcus, Jun 26 2014

Status

approved