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A065080 Denominator of Sum_{k=1..n} d(k)/k, where d() = A000005(). 2

%I #21 Aug 31 2018 09:55:13

%S 1,1,3,12,60,60,420,420,420,420,4620,4620,60060,60060,60060,240240,

%T 4084080,4084080,77597520,77597520,25865840,25865840,594914320,

%U 1784742960,8923714800,8923714800,80313433200,80313433200,2329089562800,2329089562800,72201776446800

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

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

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

%H Harry J. Smith, <a href="/A065080/b065080.txt">Table of n, a(n) for n=1..200</a>

%H Mathematics.StackExchange, <a href="https://math.stackexchange.com/questions/471326/the-asymptotic-expansion-for-the-weighted-sum-of-divisors-sum-n-leq-x-frac">The asymptotic expansion for the weighted sum of divisors</a>, Aug 19 2013.

%F 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

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

%p t:= 0:

%p for n from 1 to 50 do

%p t:= t + numtheory:-tau(n)/n;

%p A[n]:= denom(t);

%p od: seq(A[n], n=1..50); # _Robert Israel_, Mar 20 2018

%t Denominator[Accumulate[Table[DivisorSigma[0,n]/n,{n,40}]]] (* _Harvey P. Dale_, Jul 31 2016 *)

%o (PARI) { s=0; for (n=1, 200, s+=numdiv(n)/n; write("b065080.txt", n, " ", denominator(s)) ) } \\ _Harry J. Smith_, Oct 06 2009

%o (PARI) a(n) = denominator(sum(k=1, n, numdiv(k)/k)); \\ _Michel Marcus_, Mar 20 2018

%Y Cf. A000005, A060436.

%K nonn,frac

%O 1,3

%A _N. J. A. Sloane_, Nov 02 2008

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Last modified March 28 17:25 EDT 2024. Contains 371254 sequences. (Running on oeis4.)