A318320 a(n) = (psi(n) - phi(n))/2.

0, 1, 1, 2, 1, 5, 1, 4, 3, 7, 1, 10, 1, 9, 8, 8, 1, 15, 1, 14, 10, 13, 1, 20, 5, 15, 9, 18, 1, 32, 1, 16, 14, 19, 12, 30, 1, 21, 16, 28, 1, 42, 1, 26, 24, 25, 1, 40, 7, 35, 20, 30, 1, 45, 16, 36, 22, 31, 1, 64, 1, 33, 30, 32, 18, 62, 1, 38, 26, 60, 1, 60, 1, 39, 40, 42, 18, 72, 1, 56, 27, 43, 1, 84, 22, 45, 32, 52, 1, 96

Offset: 1

Author

Antti Karttunen, Aug 26 2018

Keywords

• nonn
• easy
• look

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Marcin Mazur and Bogdan V. Petrenko, Generalizations of Arnold's version of Euler's theorem for matrices, Japanese Journal of Mathematics, 5:183-189, 2010.

Crossrefs

Cf. A000010, A001615, A291784, A292786, A318325, A318326, A318442.

Differs from A069359 for the first time at n=30, where a(30) = 32, while A069359(30) = 31.

Programs

Mathematica

psi[n_] := n * Times @@ (1 + 1/FactorInteger[n][[;; , 1]]); psi[1] = 1; a[n_] := (psi[n] - EulerPhi[n])/2; Array[a, 100] (* Amiram Eldar, Dec 05 2023 *)

PARI

A318320(n) = sumdiv(n,d,(-1==moebius(n/d))*d);

PARI

A318320(n) = ((n*sumdivmult(n, d, issquarefree(d)/d))-eulerphi(n))/2;

Formula

a(n) = (A001615(n) - A000010(n))/2 = A292786(n)/2.

a(n) = A291784(n) - A000010(n).

a(n) = A318326(n) + A318442(n).

Sum_{k=1..n} a(k) = c * n^2 + O(n*log(n)), where c = 9/(4*Pi^2) = 0.227972... . - Amiram Eldar, Dec 05 2023