A074638 -id:A074638 - 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 *)

A074637 Numerator of 4 * H(n,4,3), a generalized harmonic number.

Original entry on oeis.org

4, 40, 524, 976, 20084, 491192, 13935164, 450160544, 93250876, 1249813672, 55206526972, 2657681947952, 46167204272716, 235410309457592, 14140794103168588, 14376406243883968, 978062783205294796

Offset: 1

Robert G. Wilson v, Aug 27 2002

The denominators are in A074638.

Cf. A075135.

Table[ Numerator[ Sum[1/i, {i, 3/4, n}]], {n, 1, 20}]

a(n) = numerator(4 * Sum_{j=0..n-1} 1/(4*j + 3)) = numerator(Psi(n + 3/4) - Psi(3/4)), with the Digamma function Psi(z). See a comment in A074638 with the Abramowitz-Stegun link. - Wolfdieter Lang, Apr 05 2022

Better description from T. D. Noe, Sep 04 2002

A097329 Least common multiple of {3,7,11,...,4n+3}.

Original entry on oeis.org

3, 21, 231, 1155, 21945, 504735, 4542615, 140821065, 140821065, 1830673845, 78718975335, 3699791840745, 62896461292665, 62896461292665, 3710891216267235, 3710891216267235, 248629711489904745

Offset: 0

T. D. Noe, Aug 04 2004

The first three terms are the same as the denominators in A074638.

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Cf. A004767 (4n+3), A074638 (denominator of 1/3+1/7+/11+...+1/(4n+3)).

Cf. A051539.

k:=67; [Lcm([h: h in [3..j by 4]]): j in [3..k by 4]];  // Bruno Berselli, May 03 2011

Table[LCM@@Range[3, 4n+3, 4], {n, 0, 19}]

A352395 Denominator of Sum_{k=0..n} (-1)^k / (2*k+1).

Original entry on oeis.org

1, 3, 15, 105, 315, 3465, 45045, 45045, 765765, 14549535, 14549535, 334639305, 1673196525, 5019589575, 145568097675, 4512611027925, 4512611027925, 4512611027925, 166966608033225, 166966608033225, 6845630929362225, 294362129962575675, 294362129962575675, 13835020108241056725

Offset: 0

Wolfdieter Lang, Apr 06 2022

This is not the sequence A025547(n+1)_{n>=0}, because a(32) = 1420993851085122917681925 but A025547(33) = 18472920064106597929865025. Hence it is also not the sequence A350670.
The alternating sum Sum_{k=0..n} (-1)^k/(2*k+1) = (Psi(n + 3/2) - Psi((2*n - (-1)^n)/4 + 1) - log(2) + Pi/2)/2, with the Digamma function Psi(z).
Proof by subtracting twice the negative fractions from Sum_{k=0..n} 1/(2*k+1) = A350669(n)/A350670(n) (Abramowitz-Stegun, p. 258, eq. 6.3.4.), using Sum_{j=0..k} 1/(4*j + 3) = A074637((k+1)/4)/A074638(k+1) (Abramowitz-Stegun, p. 258, eqs. 6.3.6. with 6.3.5.) and, finally, replacing in the results for the even and odd n cases the formula for Psi(3/4) = -A200134.

Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions. p.258, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. p. 258.

Cf. A007509 (numerators).

Cf. A025547, A074637, A074638, A200134, A350669, A350670.

Denominator @ Accumulate @ Table[(-1)^k/(2*k + 1), {k, 0, 25}] (* Amiram Eldar, Apr 08 2022 *)

a(n) = denominator(sum(k=0, n, (-1)^k / (2*k+1))); \\ Michel Marcus, Apr 07 2022

from fractions import Fraction
def A352395(n): return sum(Fraction(-1 if k % 2 else 1,2*k+1) for k in range(n+1)).denominator # Chai Wah Wu, May 18 2022

a(n) = denominator( (Psi(n + 3/2) - Psi((2*n - (-1)^n)/4 + 1) - log(2) + Pi/2)/2), for n >= 0, with the Digamma function. See the above comment.

a(n) = denominator(Pi/4 + (-1)^n * (Psi((n + 5/2)/2) - Psi((n + 3/2)/2))/4). - Vaclav Kotesovec, May 16 2022

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

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A074637 Numerator of 4 * H(n,4,3), a generalized harmonic number.

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A097329 Least common multiple of {3,7,11,...,4n+3}.

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A352395 Denominator of Sum_{k=0..n} (-1)^k / (2*k+1).

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