A189998 Numerator of h(n+5) - h(n) where h(n) = Sum_{k=1..n} (1/k) are the Harmonic numbers.

137, 29, 153, 743, 1879, 1627, 15797, 2021, 11899, 25381, 7793, 2627, 124877, 26987, 68879, 65003, 107699, 66167, 482897, 16167, 77293, 412561, 323959, 94781, 1323137, 255127, 587299, 504563, 255733, 145209, 2956637, 277681, 1247459, 2094661, 1558379, 433501

Offset

0,1

Comments

a(n) = Numerator of (5*n^4 + 60*n^3 + 255*n^2 + 450*n + 274)/((n+1)*(n+2)*(n+3)*(n+4)*(n+5)).

(5*n^4 + 60*n^3 + 255*n^2 + 450*n + 274)/a(n) can be factored into 2^p(n)* 3^q(n) where p(n) is a sequence of period 4 repeating [1,2,1,3] and q(n) is of period 9,repeating [0,2,2,0,1,1,0,1,1].

p(n) = A131743(n) + 1.

q(n) = A011655(n) + [0,2,2,0,0,0,0,0,0]

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Formula

a(n) = (5*n^4 + 60*n^3 + 255*n^2 + 450*n + 274)/(2^q(n)*3^(P(9,n-1) + P(9,n-2) + 1 - P(3,n))), where q(n) = (1-(-1)^n)*(3+i^(n+1))/4 + 1 and P(k,n) = floor(1/2*cos(2*n*Pi/k)+1/2).

Maple

h:= n->sum(1/k, k=1..n):seq(numer(h(n+5)-h(n)), n=0..28);
q:=n-> (1-(-1)^n)*(3+I^(n+1))/4+1:
P:=(k, n)-> floor(1/2*cos(2*n*Pi/k)+1/2):
seq( (5*n^4+60*n^3+255*n^2+450*n+274)/(2^q(n)*3^(P(9, n-1)+P(9, n-2)+1-P(3, n))), n=0..28)

Mathematica

Numerator[Table[HarmonicNumber[n+5]-HarmonicNumber[n], {n, 0, 30}]] (* Harvey P. Dale, Sep 15 2016 *)

Prog

(Magma) [137] cat [Numerator(HarmonicNumber(n+5)-HarmonicNumber(n)): n in [0..30]]; // G. C. Greubel, Jan 11 2018
(Python)
from sympy import harmonic, numer
print([numer(harmonic(n+5) - harmonic(n)) for n in range(0, 30)])
# Javier Rivera Romeu, May 22 2023

Crossrefs

Cf. A011655, A131743.

Sequence in context: A134885 A259680 A082726 * A138358 A138329 A066477

Adjacent sequences: A189995 A189996 A189997 * A189999 A190000 A190001

Keyword

nonn

Author

Gary Detlefs, May 03 2011

Status

approved