A292786 - OEIS

%I #26 Dec 05 2023 08:11:48

%S 0,2,2,4,2,10,2,8,6,14,2,20,2,18,16,16,2,30,2,28,20,26,2,40,10,30,18,

%T 36,2,64,2,32,28,38,24,60,2,42,32,56,2,84,2,52,48,50,2,80,14,70,40,60,

%U 2,90,32,72,44,62,2,128,2,66,60,64,36,124,2,76,52,120,2,120,2,78,80,84,36,144

%N a(n) = psi(n) - phi(n).

%C Even numbers that are not the terms of this sequence are 12, 102, 114, 130, ...

%H Antti Karttunen, <a href="/A292786/b292786.txt">Table of n, a(n) for n = 1..65537</a>

%F a(n) = A001615(n) - A000010(n).

%F a(n) = 2 iff n is prime.

%F a(n) = 2*A069359(n) iff n is in A070915.

%F Sum_{k=1..n} a(k) = c * n^2 + O(n*log(n)), where c =  9/(2*Pi^2) = 0.455945... (A088245). - _Amiram Eldar_, Dec 05 2023

%t psi[n_] := If[n < 1, 0, n Sum[ MoebiusMu[d]^2/d, {d, Divisors@ n}]]; Array[psi@# - EulerPhi@# &, 87] (* _Robert G. Wilson v_, Sep 23 2017 *)

%o (PARI) a001615(n) = my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1));

%o a(n) = a001615(n) - eulerphi(n); \\ after _Charles R Greathouse IV_ at A001615

%Y Cf. A000010, A001615, A051612, A069359, A070915, A088245, A291784.

%K nonn,easy,look

%O 1,2

%A _Altug Alkan_, Sep 23 2017