A189998 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

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)])