A133946 - OEIS

%I #15 Jan 15 2025 16:43:02

%S 2,2,4,2,2,6,2,2,4,8,2,6,2,2,10,2,2,6,2,8,12,2,2,6,2,2,4,12,2,12,2,2,

%T 4,2,2,16,2,2,4,8,2,14,2,2,20,2,2,6,2,8,4,2,2,6,16,12,4,2,2,12,2,2,12,

%U 2,2,20,2,2,4,8,2,16,2,2,10,2,2,22,2,8,4,2,2,18,2,2,4,2,2,22,20,2,4,2,2,6,2

%N a(n) = Sum phi(k), where the sum is over the integers k which are the "non-isolated divisors" of 2n and phi(k) is the Euler totient function (phi(k) = A000010(k)). A positive divisor k of n is non-isolated if k-1 and/ or k+1 also divides n.

%C No odd integer has any non-isolated divisors.

%C a(n) = 2n - A133945(2n).

%H Antti Karttunen, <a href="/A133946/b133946.txt">Table of n, a(n) for n = 1..32768</a>

%F (PARI) A133946(n) = { n = 2*n; sumdiv(n,d,if((d>1 && !(n%(d-1))) || !(n%(d+1)), eulerphi(d))); }; \\ _Antti Karttunen_, Jan 15 2025

%t Table[Plus @@ EulerPhi[Select[Divisors[2n], If[ # > 1, IntegerQ[2n/(# - 1)]] || IntegerQ[2n/(# + 1)] &]], {n, 1, 80}] (* _Stefan Steinerberger_, Oct 04 2007 *)

%Y Cf. A133945.

%K nonn,changed

%O 1,1

%A _Leroy Quet_, Oct 03 2007

%E More terms from _Stefan Steinerberger_, Oct 04 2007

%E Extended by _Ray Chandler_, May 28 2008