OFFSET
0,1
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
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).
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 *)
CROSSREFS
KEYWORD
nonn,easy,changed
AUTHOR
Gary Detlefs, Jul 01 2011
EXTENSIONS
Corrected and extended by Harvey P. Dale, Jul 22 2024
STATUS
approved