A181538 -id:A181538 - cp's OEIS frontend

A181540 a(n) = Sum_{k=0..n} gcd(n,k)*phi(k).

0, 1, 3, 8, 13, 26, 29, 54, 59, 84, 93, 142, 139, 202, 195, 244, 259, 352, 327, 444, 433, 508, 505, 656, 639, 776, 719, 866, 889, 1054, 1057, 1208, 1151, 1332, 1255, 1624, 1575, 1728, 1579, 1886, 2011, 2130, 2159, 2348, 2329, 2716, 2329

Offset: 0

Peter Luschny, Oct 29 2010

nonn

Row sums of triangle A181538.

Cf. A181538.

A181540 := n -> add(igcd(n,k)*numtheory[phi](k),k=0..n);

Table[Sum[GCD[n,k]EulerPhi[k],{k,0,n}],{n,0,50}] (* Harvey P. Dale, Jul 12 2020 *)

a(n) = sum(k=1, n, gcd(n, k)*eulerphi(k)); \\ Michel Marcus, May 18 2018

A181552 T(n,k) = gcd(n,k) A181549(k), triangle read by rows.

Original entry on oeis.org

1, 1, 6, 1, 3, 12, 1, 6, 4, 20, 1, 3, 4, 5, 30, 1, 6, 12, 10, 6, 72, 1, 3, 4, 5, 6, 12, 56, 1, 6, 4, 20, 6, 24, 8, 80, 1, 3, 12, 5, 6, 36, 8, 10, 99, 1, 6, 4, 10, 30, 24, 8, 20, 11, 180, 1, 3, 4, 5, 6, 12, 8, 10, 11, 18, 132, 1, 6, 12, 20, 6, 72, 8, 40, 33, 36, 12, 240

Offset: 1

Peter Luschny, Oct 30 2010

A181549(n) = sum{k|n} k mu_2(n/k) is a variant of Euler's phi function relative to the Moebius function of order 2.

			1,
1,6,
1,3,12,
1,6,.4,20,
1,3,.4,.5,30,
1,6,12,10,.6,72,
1,3,.4,.5,.6,12,56,
1,6,.4,20,.6,24,.8,80,

Peter Luschny, Sequences related to Euler's totient function.

Cf. A130212, A181538, row sums of triangle is A181553.

A181552 := (n,k) -> igcd(n,k)*A181549(k);

mu2[1] = 1; mu2[n_] := Sum[Boole[Divisible[n, d^2]]*MoebiusMu[n/d^2]*MoebiusMu[n/d], {d, Divisors[n]}]; A181549[n_] := Sum[k*mu2[n/k], {k, Divisors[n]}]; t[n_, k_] := GCD[n, k]*A181549[k]; Table[t[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Feb 05 2014 *)

cp's OEIS Frontend

A181540 a(n) = Sum_{k=0..n} gcd(n,k)*phi(k).

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