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

1,3

Comments

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

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

Formula

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

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

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A023896, A143620, A213544.

Sequence in context: A266257 A115905 A292769 * A372686 A372517 A096134

Adjacent sequences: A307994 A307995 A307996 * A307998 A307999 A308000

Keyword

nonn,look

Author

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

Status

approved