A307997 a(n) is the sum of A023896(k) over the totatives of n.

1, 1, 2, 4, 9, 11, 25, 35, 53, 52, 109, 87, 188, 174, 218, 255, 432, 301, 622, 492, 636, 633, 1109, 725, 1288, 1113, 1468, 1287, 2275, 1121, 2801, 2305, 2598, 2499, 3227, 2266, 4760, 3550, 4229, 3449, 6556, 3311, 7628, 5527, 5846, 6199, 10017, 5736, 10453, 7282, 9654, 8832, 14451, 8143, 13060

Offset: 1

Author

J. M. Bergot and Robert Israel, May 09 2019

Keywords

• nonn
• look

Comments

a(n) <= A213544(n-1) for n >= 2, with equality if and only if n is prime. - Robert Israel, May 10 2019

Examples

			a(6) = 11 because the totatives of 6, i.e. the numbers from 1 to 6 that are coprime to 6, are 1 and 5, A023896(1) = 1 and A023896(5) = 1+2+3+4=10, and 1+10=11.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Robert Israel, Plot of a(n)/n^3 for n=3 to 20000

Crossrefs

Cf. A023896, A143620, A213544.

Programs

Maple

A023896:= proc(n) option remember; convert(select(t -> igcd(t,n)=1, [$1..n]),`+`) end proc:
f:= n -> convert(map(A023896, select(t -> igcd(t,n)=1, [$1..n])),`+`):
map(f, [$1..100]);

Mathematica

A023896[n_] := If[n == 1, 1, (n/2) EulerPhi[n]];
a[n_] := Sum[Boole[GCD[n, k] == 1] A023896[k], {k, 1, n}];
Array[a, 100] (* Jean-François Alcover, Jul 31 2020 *)

PARI

s(n) = if(n<2, n>0, n*eulerphi(n)/2); \\ A023896
a(n) = sum(k=1, n, if (gcd(n,k)==1, s(k))); \\ Michel Marcus, May 10 2019

Formula

a(n) = Sum_{1<=k<=n; gcd(k,n)=1} A023896(k).

a(n) = Sum_{k=1..n} k*A143620(n,k).