|
%I #27 Dec 05 2023 03:04:58
%S 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,
%T 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,
%U 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
%N a(n) = (psi(n) - phi(n))/2.
%H Antti Karttunen, <a href="/A318320/b318320.txt">Table of n, a(n) for n = 1..65537</a>
%H Marcin Mazur and Bogdan V. Petrenko, <a href="http://people.math.binghamton.edu/mazur/papers/pub5.pdf">Generalizations of Arnold's version of Euler's theorem for matrices</a>, Japanese Journal of Mathematics, 5:183-189, 2010.
%F a(n) = (A001615(n) - A000010(n))/2 = A292786(n)/2.
%F a(n) = A291784(n) - A000010(n).
%F a(n) = A318326(n) + A318442(n).
%F 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
%t 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 *)
%o (PARI) A318320(n) = sumdiv(n,d,(-1==moebius(n/d))*d);
%o (PARI) A318320(n) = ((n*sumdivmult(n, d, issquarefree(d)/d))-eulerphi(n))/2;
%Y Cf. A000010, A001615, A291784, A292786, A318325, A318326, A318442.
%Y Differs from A069359 for the first time at n=30, where a(30) = 32, while A069359(30) = 31.
%K nonn,easy,look
%O 1,4
%A _Antti Karttunen_, Aug 26 2018
|
