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A189998
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Numerator of h(n+5) - h(n) where h(n) = Sum_{k=1..n} (1/k) are the Harmonic numbers.
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3
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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
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OFFSET
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0,1
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COMMENTS
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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].
q(n) = A011655(n) + [0,2,2,0,0,0,0,0,0]
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LINKS
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FORMULA
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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).
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MAPLE
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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)
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MATHEMATICA
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Numerator[Table[HarmonicNumber[n+5]-HarmonicNumber[n], {n, 0, 30}]] (* Harvey P. Dale, Sep 15 2016 *)
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PROG
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(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)])
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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