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A060436 Numerator of Sum_{k=1..n} d(k)/k, where d() = A000005(). 2

%I

%S 1,2,8,41,229,269,2003,2213,2353,2521,28571,30881,410693,427853,

%T 443869,1850551,31939847,33301207,640891093,664170349,226316943,

%U 231019823,5365187609,16690477147,84523231511,85896110711,784963282799,802173304199,23423652688171

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

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

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

%H Robert Israel, <a href="/A060436/b060436.txt">Table of n, a(n) for n = 1..2296</a>

%H Vaclav Kotesovec, <a href="/A060436/a060436.jpg">Graph - The asymptotic ratio of Sum_{k=1..n} d(k)/k</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 = 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

%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]:= numer(t);

%p od:

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

%t 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]

%Y Cf. A000005, A065080.

%K nonn,frac

%O 1,2

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

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Last modified September 18 21:53 EDT 2020. Contains 337173 sequences. (Running on oeis4.)