A074596 -id:A074596 - cp's OEIS frontend

A075135 Numerator of the generalized harmonic number H(n,3,1) described below.

1, 5, 39, 209, 2857, 11883, 233057, 2632787, 13468239, 13739939, 433545709, 7488194853, 281072414761, 284780929571, 12393920563953, 288249495707519, 2038704876507433, 2058454144222533, 2077126179153173, 60750140156034617

Offset: 1

T. D. Noe, Sep 04 2002

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).
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.
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.
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).
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).

			a(3)=39 because 1 + 1/4 + 1/7 = 39/28.

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 263, 269, 272, 297, 302, 356.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 4th ed., Oxford Univ. Press, 1971, page 88.

Eric Weisstein's World of Mathematics, Harmonic Series
Eric Weisstein's World of Mathematics, Harmonic Number
Eric Weisstein's World of Mathematics, Jeep Problem

Cf. A001008, A002805, A025550, A025547, A051536, A051539, A074637, A074638, A075136-A075144.
Cf. A051540, A074596, A074597, A074598.
Cf. A007406, A007407.

a=3; b=1; maxN=20; s=0; Numerator[Table[s+=1/(a n + b), {n, 0, maxN-1}]]
Accumulate[1/Range[1,60,3]]//Numerator (* Harvey P. Dale, Dec 30 2019 *)

A051536 a(n) = least common multiple of {1, 4, 7, 10, 13 ..., 3n+1} (A016777).

Original entry on oeis.org

1, 4, 28, 140, 1820, 7280, 138320, 1521520, 7607600, 7607600, 235835600, 4009205200, 148340592400, 148340592400, 6378645473200, 146708845883600, 1026961921185200, 1026961921185200, 1026961921185200, 29781895714370800

Offset: 0

Asher Auel

This sequence coincides with the sequence of denominators of 1 + 1/4 + 1/7 + 1/10 + ... + 1/(3*n + 1) for n < 29. - T. D. Noe, Aug 04 2004

The sequence coincides with the sequence of denominators of 1 - 1/4 + 1/7 - 1/10 + ... + (-1)^n/(3*n + 1) for n < 45. - Peter Bala, Feb 19 2024

			a(4) = lcm of {1, 4, 7, 10, 13} = 1820.

Cf. A016777.
The numerators are in A074596.
Cf. A075135, A059457, A051540.

a051536 n = a051536_list !! (n-1)
a051536_list = scanl1 lcm a016777_list
-- Reinhard Zumkeller, Feb 28 2013, Feb 10 2012

k:=58; [Lcm([h: h in [1..j by 3]]): j in [1..k by 3]]; // Bruno Berselli, May 03 2011

Table[ Denominator[ Sum[1/i, {i, 1/3, n}]], {n, 1, 20}]
Table[ Apply[ LCM, Join[{1}, Table[1 + 3i, {i, 0, n}]]], {n, 0, 19}]
Table[Denominator[Total[1/Range[1, 3n+1, 3]]], {n, 0, 29}]
Module[{nn=30,lst},lst=3*Range[0,nn]+1;Table[LCM@@Take[lst,n],{n,nn}]] (* Harvey P. Dale, Sep 30 2012 *)

a(n)=lcm(vector(n,i,3*i+1)) \\ Charles R Greathouse IV, Feb 09 2017

Edited by Robert G. Wilson v, Aug 27 2002

cp's OEIS Frontend

A075135 Numerator of the generalized harmonic number H(n,3,1) described below.

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