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A192449 Numerator of h(n+7) - h(n), where h(n) = Sum_{k=1..n} 1/k.

%I A192449 #23 Oct 23 2024 01:15:19
%S A192449 363,481,3349,2761,25961,22727,263111,237371,21635,8837,695089,529331,
%T A192449 9407549,679829,641069,6671911,36404897,4075097,2159257,1412139,
%U A192449 36516143,35036093,88771727,3715069
%N A192449 Numerator of h(n+7) - h(n), where h(n) = Sum_{k=1..n} 1/k.
%C A192449 a(n) = numerator((7*n^6 + 168*n^5 + 1610*n^4 + 7840*n^3 + 20307*n^2 + 26264*n + 13068)/((n+1)*(n+2)*...*(n+7)));
%C A192449 (7*n^6 + 168*n^5 + 1610*n^4 + 7840*n^3 + 20307*n^2 + 26264*n + 13068)/a(n) can be factored into 2^m(n)*3^p(n)*5^(q1(n) + q2(n)) where
%C A192449 m(n) is of period 4, repeating [2,4,3,4]
%C A192449 p(n) is of period 9, repeating [2,2,2,1,1,2,1,1,2]
%C A192449 q1(n) is of period 5, repeating [0,0,0,1,1]
%C A192449 q2(n) is of period 25, repeating [0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]
%H A192449 Harvey P. Dale, <a href="/A192449/b192449.txt">Table of n, a(n) for n = 0..1000</a>
%F A192449 a(n) = (7*n^6 + 168*n^5 + 1610*n^4 + 7840*n^3 + 20307*n^2 + 26264*n + 13068)/ (2^m(n)*3^p(n)*5^q(n)) where
%F A192449   m(n) = P(1,4,3,n) + 2*P(0,2,1,n) + 2,
%F A192449   p(n) = P(0,3,2,n) + P(7,9,7,n) + 1,
%F A192449   q(n) = P(0,5,3,n) + P(15,15,23,n),
%F A192449   P(x,y,z,n) = floor(((n+x) mod y)/z).
%p A192449 h:= n-> sum(1/k,k=1..n):seq(numer(h(n+7)-h(n)), n=0..23);
%p A192449 P:=(x,y,z,n)-> floor(((n+x) mod y)/z):
%p A192449 m:=n-> P(1,4,3,n)+2*P(0,2,1,n)+2:
%p A192449 p:=n-> P(0,3,2,n)+P(7,9,7,n)+1:
%p A192449 q:=n-> P(0,5,3,n)+P(15,15,23,n):
%p A192449 N7:=n->(7*n^6+168*n^5+1610*n^4+7840*n^3+20307*n^2+26264*n+13068): seq(N7(n)/(2^m(n)*3^p(n)*5^q(n)), n=0..23);
%p A192449 # Alternative implementation, _R. J. Mathar_, Jul 12 2011:
%p A192449 A192449 := proc(n) add(1/i,i=n+1..n+7) ; numer(%) ; end proc:
%t A192449 #[[8]]-#[[1]]&/@Partition[HarmonicNumber[Range[0,30]],8,1]//Numerator (* _Harvey P. Dale_, Jul 22 2024 *)
%Y A192449 Cf. A188386, A189642, A189998, A192359.
%K A192449 nonn,easy
%O A192449 0,1
%A A192449 _Gary Detlefs_, Jul 01 2011
%E A192449 Corrected and extended by _Harvey P. Dale_, Jul 22 2024