A192359 - OEIS

%I #32 Sep 08 2022 08:45:57

%S 49,223,341,2509,2131,20417,18107,30233,96163,1959,36177,51939,436511,

%T 598433,80507,532541,1388179,1785181,378013,95003,1181909,4370849,

%U 2671363,3240049,1560647,9333997,5547947,2185691,5138581,1201967,10493071,12159157,28060691,32250013

%N Numerator of h(n+6) - h(n), where h(n) = Sum_{k=1..n} 1/k.

%C Numerator of (2*n+7)*(3*n^4 + 42*n^3 + 203*n^2 + 392*n + 252)/((n+1)*(n+2)*...*(n+6)).

%C (2*n+7)*(3*n^4 + 42*n^3 + 203*n^2 + 392*n + 252)/a(n) can be factored into 2^m(n)*3^p(n)*5^(q1(n) + q2(n)) where

%C m(n) is of period 4, repeating [2,2,3,3]

%C p(n) is of period 9, repeating [2,2,2,1,1,1,1,1,1]

%C q1(n) is of period 5, repeating [0,0,0,0,1]

%C q2(n) is of period 25, repeating [0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0].

%H Robert Israel, <a href="/A192359/b192359.txt">Table of n, a(n) for n = 0..10000</a>

%F a(n) = (2*n+7)*(3*n^4 + 42*n^3 + 203*n^2 + 392*n + 252)/(2^(P(0,4,2,n)+2) * 3^(P(6,9,6,n)+1)*5^(P(0,5,4,n)+P(15,25,24,n))), where P(x,y,z,n) = floor(((n+x)mod y)/z).

%p h:= n-> sum(1/k,k=1..n):seq(numer(h(n+6)-h(n)), n=0..33);

%p P:=(x,y,z,n)-> floor(((n+x)mod y)/z):

%p a:=n->(2*n+7)*(3*n^4+42*n^3+203*n^2+392*n+252)/(2^(P(0,4,2,n)+2)*3^(P(6,9,6,n)+1)*5^(P(0,5,4,n)+P(15,25,24,n))):

%p seq(a(n), n=0..25);

%t Numerator[Table[HarmonicNumber[n+6]-HarmonicNumber[n],{n,0,40}]] (* _Harvey P. Dale_, Mar 27 2015 *)

%o (PARI) h(n) = sum(k=1, n, 1/k);

%o a(n) = numerator(h(n+6)-h(n)); \\ _Michel Marcus_, Apr 15 2017

%o (Magma) [49] cat [Numerator(HarmonicNumber(n+6) - HarmonicNumber(n)): n in [1..40]]; // _G. C. Greubel_, Oct 20 2018

%o (GAP) List(List([0..35],n->Sum([1..n+6],k->(1/k))-Sum([1..n],k->(1/k))),NumeratorRat); # _Muniru A Asiru_, Oct 21 2018

%Y Cf. A188386, A189642, A189998, A021913.

%K nonn,frac,look

%O 0,1

%A _Gary Detlefs_, Jun 28 2011