A183002 - OEIS

%I #27 Jan 19 2024 04:57:34

%S 0,0,0,2,2,4,4,6,8,10,10,14,14,16,18,22,22,26,26,30,32,34,34,40,42,44,

%T 46,50,50,56,56,60,62,64,66,74,74,76,78,84,84,90,90,94,98,100,100,108,

%U 110,114,116,120,120,126,128,134,136,138,138,148,148,150,154

%N a(n) is the total number of noncentral divisors in all positive integers <= n.

%C The original name was: Partial sums of A161840.

%F a(n) = Sum_{k=1..n} (tau(k)-2 + (tau(k) mod 2)), tau = A000005.

%F a(n) ~ n * (log(n) + 2*gamma - 3), where gamma is Euler's constant (A001620). - _Amiram Eldar_, Jan 19 2024

%p with(numtheory):

%p b:= n-> (x-> x-2+(x mod 2))(tau(n)):

%p a:= proc(n) option remember; b(n) +`if`(n=1, 0, a(n-1)) end:

%p seq(a(n), n=1..100);

%t Accumulate[Table[d = DivisorSigma[0, n]; If[OddQ[d], d - 1, d - 2], {n, 100}]]

%o (PARI) lista(nmax) = {my(s = 0, d); for(n = 1, nmax, d = numdiv(n); s += (d + d%2 - 2); print1(s, ", ")); } \\ _Amiram Eldar_, Jan 19 2024

%Y Cf. A000005, A001620, A161840, A183003.

%K nonn,easy

%O 1,4

%A _Omar E. Pol_, Jan 27 2011

%E New name from _Omar E. Pol_, Jan 04 2022