A230583 a(n) = floor(s(n) - n*(log(n) + 2*Gamma - 1)), where s(n) = sum_{k=1..n} tau(k), where tau(k) is the number of divisors of k.

0, 1, 1, 1, 1, 2, 1, 2, 1, 2, 0, 3, 1, 1, 2, 3, 1, 3, 1, 2, 2, 2, 0, 4, 2, 2, 1, 3, 0, 4, 1, 3, 2, 1, 1, 5, 2, 1, 1, 4, 1, 4, 1, 2, 3, 2, -1, 4, 2, 3, 2, 3, 0, 3, 2, 4, 3, 2, -1, 6, 2, 1, 2, 3, 2, 5, 1, 2, 1, 3, 0, 6, 3, 2, 2, 3, 1, 4, 0, 5, 4, 2, -1, 5, 4, 2

Offset

1,6

Comments

Dirichlet proved this sequence is O(n^1/2).

Links

G. C. Greubel, Table of n, a(n) for n = 1..10000

Wikipedia, Divisor summatory function

Wikipedia, Euler-Mascheroni constant

Formula

a(n) = Sum(A000005(k), k=1..n) - n*(log(n) + 0.1544313298), where the decimal is (approximately) 2*Gamma-1.

Mathematica

s = 0; Table[s = s + DivisorSigma[0, n]; Floor[s - n*(Log[n] + 2*EulerGamma - 1)], {n, 100}] (* T. D. Noe, Nov 04 2013 *)

Prog

(JavaScript)
function sigma(n, k) {
var j, s, sn;
s=0;
sn=Math.sqrt(n);
for (j=1; j<sn; j++) if (n%j==0) s+=Math.pow(j, k)+Math.pow(n/j, k);
if (n%sn==0) s+=Math.pow(sn, k);
return s;
}
c=0;
for (i=1; i<100; i++) {
c+=sigma(i, 0);
document.write(Math.floor(c-i*(Math.log(i)+0.1544313298))+", ");
}

Crossrefs

Cf. A000005, A001620, A006218, A230501.

Sequence in context: A316828 A350223 A274820 * A197366 A245715 A337736

Adjacent sequences: A230580 A230581 A230582 * A230584 A230585 A230586

Keyword

sign

Author

Jon Perry, Oct 24 2013

Status

approved