A117295 - OEIS

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A117295

a(n) = phi(n)*(n - phi(n)).

0, 1, 2, 4, 4, 8, 6, 16, 18, 24, 10, 32, 12, 48, 56, 64, 16, 72, 18, 96, 108, 120, 22, 128, 100, 168, 162, 192, 28, 176, 30, 256, 260, 288, 264, 288, 36, 360, 360, 384, 40, 360, 42, 480, 504, 528, 46, 512, 294, 600, 608, 672, 52, 648, 600, 768, 756, 840, 58, 704, 60

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

LINKS

Table of n, a(n) for n=1..61.

C. A. Nicol, On restricted partitions and a generalization of the Euler phi number and the Moebius function, PNAS September 1, 1953 vol. 39 no. 9 963-968.

FORMULA

For n > 1, a(n) = Sum_{k=1..n-1} PHI(k,n)^2 where PHI(k,n) = phi(n)*mu(n/GCD(k,n))/phi(n/GCD(k,n)), and has been considered by C. Nicol under the name G(n). - Michel Marcus, Nov 11 2012

From Amiram Eldar, Dec 21 2023: (Start)

a(n) = A002618(n) - A127473(n).

Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = A059956 - A065464 = 0.179677... . (End)

EXAMPLE

a(8) = phi(8)*(8 - phi(8)) = 4*4 = 16.

MATHEMATICA

a[n_] := Module[{phi = EulerPhi[n]}, phi*(n - phi)]; Array[a, 100] (* Amiram Eldar, Dec 21 2023 *)

PROG

(PARI) a(n) = eulerphi(n)*(n-eulerphi(n));

CROSSREFS

Cf. A000010, A002618, A051953, A127473.

Cf. A059956, A065464.