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A230583 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
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,6

COMMENTS

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

LINKS

Table of n, a(n) for n=1..86.

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: A029445 A274920 A274820 * A197366 A245715 A047885

Adjacent sequences:  A230580 A230581 A230582 * A230584 A230585 A230586

KEYWORD

sign

AUTHOR

Jon Perry, Oct 24 2013

STATUS

approved

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Last modified February 17 18:31 EST 2018. Contains 299296 sequences. (Running on oeis4.)