A189998 -id:A189998 - cp's OEIS frontend

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

Gary Detlefs, Jun 28 2011

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].

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

Cf. A188386, A189642, A189998, A021913.

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

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

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);

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

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

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).

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

Original entry on oeis.org

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

Gary Detlefs, Jul 01 2011

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]

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

Cf. A188386, A189642, A189998, A192359.

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:

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

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).

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

cp's OEIS Frontend

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

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