%I A075135
%S A075135 1,5,39,209,2857,11883,233057,2632787,13468239,13739939,433545709,
%T A075135 7488194853,281072414761,284780929571,12393920563953,288249495707519,
%U A075135 2038704876507433,2058454144222533,2077126179153173,60750140156034617
%N A075135 Numerator of the generalized harmonic number H(n,3,1) described below.
%C A075135 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 A075135 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 A075135 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 A075135 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 A075135 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 A075135 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, 
               Reading, MA, 1990, p. 263, 269, 272, 297, 302, 356.
%D A075135 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 
               4th ed., Oxford Univ. Press, 1971, page 88.
%H A075135 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               HarmonicSeries.html">Harmonic Series</a>
%H A075135 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               HarmonicNumber.html">Harmonic Number</a>
%H A075135 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               JeepProblem.html">Jeep Problem</a>
%e A075135 a(3)=39 because 1 + 1/4 + 1/7 = 39/28.
%t A075135 a=3; b=1; maxN=20; s=0; Numerator[Table[s+=1/(a n + b), {n, 0, maxN-1}]]
%Y A075135 Cf. A001008, A002805, A025550, A025547, A051536, A051539, A074637, A074638, 
               A075136-A075144.
%Y A075135 Cf. A051540, A074596, A074597, A074598.
%Y A075135 Cf. A007406, A007407.
%Y A075135 Sequence in context: A153267 A064445 A123614 this_sequence A053573 A003482 
               A135849
%Y A075135 Adjacent sequences: A075132 A075133 A075134 this_sequence A075136 A075137 
               A075138
%K A075135 easy,frac,nonn
%O A075135 1,2
%A A075135 T. D. Noe (noe(AT)sspectra.com), Sep 04 2002