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%I #7 Dec 30 2019 15:15:36
%S 1,5,39,209,2857,11883,233057,2632787,13468239,13739939,433545709,
%T 7488194853,281072414761,284780929571,12393920563953,288249495707519,
%U 2038704876507433,2058454144222533,2077126179153173,60750140156034617
%N Numerator of the generalized harmonic number H(n,3,1) described below.
%C For integers a and b, H(n,a,b) is the sum of the fractions 1/(a i + b), i = 0,1,..,n-1. This database already contains six instances of generalized harmonic numbers. Partial sums of the harmonic series 1+1/2+1/3+1/4+... are given by the sequence of harmonic numbers H(n,1,1) = A001008(n) / A002805(n).
%C The Jeep problem gives rise to the series H(n,2,1) = A025550(n) / A025547(n). Recent additions to the database are 3 * H(n,3,1) = A074596(n) / A051536(n), 3 * H(n,3,2) = A074597(n) / A051540(n), 4 * H(n,4,1) = A074598(n) / A051539(n) and 4 * H(n,4,3) = A074637(n) / A074638(n) . The numerator of H(n,4,1) is A075136. The fractions H(n,5,1), H(n,5,2), H(n,5,3) and H(n,5,4) are in A075137-A075144.
%C The sequence H(n,a,b) is of interest only when a and b are relatively prime. The sequence can also be computed as H(n,a,b) = (PolyGamma[n+1+b/a] - PolyGamma[1+b/a])/a. The sequence H(n,a,b) diverges for all a and b.
%C According to Hardy and Wright, if p is an odd prime, then p divides the numerator of the harmonic number H(p-1,1,1). This result can be extended to generalized harmonic numbers: for odd integer n, let q = (n-2)a + 2b. If q is prime, then q divides the numerator of H(n-1,a,b). For this sequence (a=3, b=1) we conclude that 11 divides a(4), 17 divides a(6), 29 divides a(10) and 47 divides a(16).
%C Graham, Knuth and Patashnik define another type of generalized harmonic number as the sum of fractions 1/i^k, i=1,...,n. For k=2, the sequence of fractions is A007406(n) / A007407(n).
%D R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 263, 269, 272, 297, 302, 356.
%D G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 4th ed., Oxford Univ. Press, 1971, page 88.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HarmonicSeries.html">Harmonic Series</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HarmonicNumber.html">Harmonic Number</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/JeepProblem.html">Jeep Problem</a>
%e a(3)=39 because 1 + 1/4 + 1/7 = 39/28.
%t a=3; b=1; maxN=20; s=0; Numerator[Table[s+=1/(a n + b), {n, 0, maxN-1}]]
%t Accumulate[1/Range[1,60,3]]//Numerator (* _Harvey P. Dale_, Dec 30 2019 *)
%Y Cf. A001008, A002805, A025550, A025547, A051536, A051539, A074637, A074638, A075136-A075144.
%Y Cf. A051540, A074596, A074597, A074598.
%Y Cf. A007406, A007407.
%K easy,frac,nonn
%O 1,2
%A _T. D. Noe_, Sep 04 2002
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