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

363, 481, 3349, 2761, 25961, 22727, 263111, 237371, 21635, 8837, 695089, 529331, 9407549, 679829, 641069, 6671911, 36404897, 4075097, 2159257, 1412139, 36516143, 35036093, 88771727, 3715069

Offset: 0

Author

Gary Detlefs, Jul 01 2011

Keywords

• nonn
• easy

Comments

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)));

(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

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

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

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

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]

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A188386, A189642, A189998, A192359.

Programs

Maple

h:= n-> sum(1/k,k=1..n):seq(numer(h(n+7)-h(n)), n=0..23);
P:=(x,y,z,n)-> floor(((n+x) mod y)/z):
m:=n-> P(1,4,3,n)+2*P(0,2,1,n)+2:
p:=n-> P(0,3,2,n)+P(7,9,7,n)+1:
q:=n-> P(0,5,3,n)+P(15,15,23,n):
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);
# Alternative implementation, R. J. Mathar, Jul 12 2011:
A192449 := proc(n) add(1/i,i=n+1..n+7) ; numer(%) ; end proc:

Mathematica

#[[8]]-#[[1]]&/@Partition[HarmonicNumber[Range[0,30]],8,1]//Numerator (* Harvey P. Dale, Jul 22 2024 *)

Formula

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

m(n) = P(1,4,3,n) + 2*P(0,2,1,n) + 2,

p(n) = P(0,3,2,n) + P(7,9,7,n) + 1,

q(n) = P(0,5,3,n) + P(15,15,23,n),

P(x,y,z,n) = floor(((n+x) mod y)/z).

Extensions

Corrected and extended by Harvey P. Dale, Jul 22 2024