A060436 Numerator of Sum_{k=1..n} d(k)/k, where d() = A000005().

1, 2, 8, 41, 229, 269, 2003, 2213, 2353, 2521, 28571, 30881, 410693, 427853, 443869, 1850551, 31939847, 33301207, 640891093, 664170349, 226316943, 231019823, 5365187609, 16690477147, 84523231511, 85896110711, 784963282799, 802173304199, 23423652688171

Offset

1,2

Comments

The old entry with this sequence number was a duplicate of A054845.

References

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

Links

Robert Israel, Table of n, a(n) for n = 1..2296

Vaclav Kotesovec, Graph - The asymptotic ratio of Sum_{k=1..n} d(k)/k

Mathematics.StackExchange, The asymptotic expansion for the weighted sum of divisors, Aug 19 2013.

Formula

Sum_{k=1..n} A000005(k)/k = a(n)/A065080(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

Example

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

Maple

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

Mathematica

l = {}; For[n = 0, n <= 1000, n++, c = 0; If[PrimeQ[n], c = c + 1]; For[k = 1, Prime[k] <= n/2, k++, For[j = 0, Prime[k + j] <= n, j++, If[Sum[Prime[i], {i, k, k + j}] == n, c = c + 1] ] ] AppendTo[l, c] ]; l [From Jake Foster, Oct 27 2008]

Crossrefs

Cf. A000005, A065080.

Sequence in context: A337753 A340992 A348474 * A020083 A333093 A217362

Adjacent sequences: A060433 A060434 A060435 * A060437 A060438 A060439

Keyword

nonn,frac

Author

N. J. A. Sloane, Nov 02 2008

Status

approved