A065080 Denominator of Sum_{k=1..n} d(k)/k, where d() = A000005().

1, 1, 3, 12, 60, 60, 420, 420, 420, 420, 4620, 4620, 60060, 60060, 60060, 240240, 4084080, 4084080, 77597520, 77597520, 25865840, 25865840, 594914320, 1784742960, 8923714800, 8923714800, 80313433200, 80313433200, 2329089562800, 2329089562800, 72201776446800

Offset: 1

Author

N. J. A. Sloane, Nov 02 2008

Keywords

• nonn
• frac

Comments

The old entry with this sequence number was a duplicate of A034093.

Examples

			1, 2, 8/3, 41/12, 229/60, 269/60, 2003/420, 2213/420, 2353/420, 2521/420, 28571/4620, 30881/4620, ...
		

References

• M. N. Huxley, Area, Lattice Points and Exponential Sums, Oxford, 1996; p. 237.

Links

• Harry J. Smith, Table of n, a(n) for n=1..200
• Mathematics.StackExchange, The asymptotic expansion for the weighted sum of divisors, Aug 19 2013.

Crossrefs

Cf. A000005, A060436.

Programs

Maple

t:= 0:
for n from 1 to 50 do
  t:= t + numtheory:-tau(n)/n;
  A[n]:= denom(t);
od: seq(A[n], n=1..50); # Robert Israel, Mar 20 2018

Mathematica

Denominator[Accumulate[Table[DivisorSigma[0,n]/n,{n,40}]]] (* Harvey P. Dale, Jul 31 2016 *)

PARI

{ s=0; for (n=1, 200, s+=numdiv(n)/n; write("b065080.txt", n, " ", denominator(s)) ) } \\ Harry J. Smith, Oct 06 2009

PARI

a(n) = denominator(sum(k=1, n, numdiv(k)/k)); \\ Michel Marcus, Mar 20 2018

Formula

Sum_{k=1..n} A000005(k)/k = A060436(n)/a(n) ~ log(n)^2/2 + 2*gamma*log(n) + gamma^2 - 2*gamma_1, where gamma is the Euler-Mascheroni constant A001620 and gamma_1 is the first Stieltjes constant A082633. - Vaclav Kotesovec, Aug 30 2018