A192359 Numerator of h(n+6) - h(n), where h(n) = Sum_{k=1..n} 1/k.

49, 223, 341, 2509, 2131, 20417, 18107, 30233, 96163, 1959, 36177, 51939, 436511, 598433, 80507, 532541, 1388179, 1785181, 378013, 95003, 1181909, 4370849, 2671363, 3240049, 1560647, 9333997, 5547947, 2185691, 5138581, 1201967, 10493071, 12159157, 28060691, 32250013

Offset: 0

Author

Gary Detlefs, Jun 28 2011

Keywords

• nonn
• frac
• look

Comments

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

(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

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

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

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

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].

Links

• Robert Israel, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A188386, A189642, A189998, A021913.

Programs

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

Magma

[49] cat [Numerator(HarmonicNumber(n+6) - HarmonicNumber(n)): n in [1..40]]; // G. C. Greubel, Oct 20 2018

Maple

h:= n-> sum(1/k,k=1..n):seq(numer(h(n+6)-h(n)), n=0..33);
P:=(x,y,z,n)-> floor(((n+x)mod y)/z):
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))):
seq(a(n), n=0..25);

Mathematica

Numerator[Table[HarmonicNumber[n+6]-HarmonicNumber[n],{n,0,40}]] (* Harvey P. Dale, Mar 27 2015 *)

PARI

h(n) = sum(k=1, n, 1/k);
a(n) = numerator(h(n+6)-h(n)); \\ Michel Marcus, Apr 15 2017

Formula

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).