A183002 a(n) is the total number of noncentral divisors in all positive integers <= n.

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, 46, 50, 50, 56, 56, 60, 62, 64, 66, 74, 74, 76, 78, 84, 84, 90, 90, 94, 98, 100, 100, 108, 110, 114, 116, 120, 120, 126, 128, 134, 136, 138, 138, 148, 148, 150, 154

Offset

1,4

Comments

The original name was: Partial sums of A161840.

Links

Table of n, a(n) for n=1..63.

Formula

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

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

Maple

with(numtheory):
b:= n-> (x-> x-2+(x mod 2))(tau(n)):
a:= proc(n) option remember; b(n) +`if`(n=1, 0, a(n-1)) end:
seq(a(n), n=1..100);

Mathematica

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

Prog

(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

Crossrefs

Cf. A000005, A001620, A161840, A183003.

Sequence in context: A352227 A334001 A341731 * A211859 A057601 A294150

Adjacent sequences: A182999 A183000 A183001 * A183003 A183004 A183005

Keyword

nonn,easy

Author

Omar E. Pol, Jan 27 2011

Extensions

New name from Omar E. Pol, Jan 04 2022

Status

approved