A294964 - OEIS

%I #15 Sep 08 2022 08:46:20

%S 1,27,1487,71207,423323,5021921,208393341,19767960169,9496615779853,

%T 112702096556215,7360072449683999,524616965933727859,

%U 526363371877036219,43813027890740553917,781806518388353706041,148866078528885256002173,15064339628673236669081953,538212602352090865654383697

%N Numerators of the partial sums of the reciprocals of the positive numbers (k + 1)*(6*k + 5) = A049452(k+1).

%C The corresponding denominators are given in A294965.

%C For the general case V(m,r;n) = Sum_{k=0..n} 1/((k + 1)*(m*k + r)) = (1/(m - r))*Sum_{k=0..n} (m/(m*k + r) - 1/(k+1)), for r = 1, ..., m-1 and m = 2, 3, ..., and their limits see a comment in A294512. Here [m,r] = [6,5].

%C The limit of the series is V(6,5) = lim_{n -> oo} V(6,5;n) = . The value is (3/2)*log(3) + 2*log(2) - (1/2)*Pi*sqrt(3) = 0.3135137477... given in A294966.

%D Max Koecher, Klassische elementare Analysis, Birkhäuser, Basel, Boston, 1987, Eulersche Reihen, pp. 189 - 193.

%H Robert Israel, <a href="/A294964/b294964.txt">Table of n, a(n) for n = 0..640</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DigammaFunction.html">Digamma Function</a>

%F a(n) = numerator(V(6,5;n)) with V(6,5;n) = Sum_{k=0..n} 1/((k + 1)*(6*k + 5)) = Sum_{k=0..n} 1/A049452(k+1) = Sum_{k=0..n} (1/(k + 5/6) - 1/(k + 1)) = -Psi(5/6) + Psi(n+11/6) - (gamma + Psi(n+2)) with the digamma function Psi and the Euler-Mascheroni constant gamma = -Psi(1) from A001620.

%e The rationals V(6,5;n), n >= 0, begin: 1/5, 27/110, 1487/5610, 71207/258060, 423323/1496748, 5021921/17462060, 208393341/715944460, 19767960169/67298779240, 9496615779853/32101517697480, ...

%e V(6,5;10^6) = 0.313513577 (Maple, 10 digits) to be compared with the rounded ten digits 0.3135137478 obtained from V(6,5) given in A294966.

%p map(numer,  ListTools:-PartialSums([seq(1/(k+1)/(6*k+5),k=0..20)])); # _Robert Israel_, Nov 29 2017

%t Table[Numerator[Sum[1/((k+1)*(6*k+5)), {k,0,n}]], {n,0,25}] (* _G. C. Greubel_, Aug 29 2018 *)

%o (PARI) a(n) = numerator(sum(k=0, n, 1/((k + 1)*(6*k + 5)))); \\ _Michel Marcus_, Nov 27 2017

%o (Magma) [Numerator((&+[1/((k+1)*(6*k+5)): k in [0..n]])): n in [0..25]]; // _G. C. Greubel_, Aug 29 2018

%Y Cf. A001620, A049452, A294512, A294966

%K nonn,frac,easy

%O 0,2

%A _Wolfdieter Lang_, Nov 27 2017