cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A074598 Numerator of 4 * H(n,4,1), a generalized harmonic number. See A075136.

Original entry on oeis.org

4, 24, 236, 3248, 57556, 416152, 2136452, 63349408, 710302388, 26725332056, 1112171931196, 375714836272, 2662087948804, 142662781936712, 2738366988282628, 168623511779891008, 170109214167178588
Offset: 1

Views

Author

Robert G. Wilson v, Aug 27 2002

Keywords

Crossrefs

The denominators are in A051539.
Cf. A075136.

Programs

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

A089153 Duplicate of A075136.

Original entry on oeis.org

1, 6, 59, 812, 14389, 104038, 534113, 15837352, 177575597, 6681333014
Offset: 1

Views

Author

Keywords

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

Original entry on oeis.org

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

Views

Author

T. D. Noe, Sep 04 2002

Keywords

Comments

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).

Examples

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

References

  • 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.

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)

A051539 a(n) is the least common multiple of {1, 5, 9, 13, 17, ..., 4n+1} (A016813).

Original entry on oeis.org

1, 5, 45, 585, 9945, 69615, 348075, 10094175, 111035925, 4108329225, 168441498225, 168441498225, 1179090487575, 62491795841475, 1187344120988025, 72427991380269525, 72427991380269525, 1665843801746199075, 121606597527472532475, 121606597527472532475
Offset: 0

Views

Author

Asher Auel

Keywords

Comments

This sequence coincides with sequence A097328 of denominators of 1 + 1/5 + 1/9 + 1/13 + ... + 1/(4n+1) iff n < 12. - T. D. Noe, Aug 04 2004

Examples

			a(3) = LCM {1, 5, 9, 13} = 585.

Links

Crossrefs

Cf. A016813, A075136, A074598, A097329.

Programs

  • Magma
    k:=77; [Lcm([h: h in [1..j by 4]]): j in [1..k by 4]];  // Bruno Berselli, May 03 2011
  • Mathematica
    Table[ Apply[ LCM, Join[{1}, Table[1 + 4i, {i, 0, n}]]], {n, 0, 19}]
nn=20;Table[LCM@@Take[4Range[0,nn-1]+1,n],{n,nn}] (* Harvey P. Dale, Jul 04 2011 *)
Table[LCM@@NestList[#+4&,1,n],{n,0,20}] (* Harvey P. Dale, Nov 21 2016 *)
  • PARI
    a(n)=lcm(vector(n,i,4*n+1)) \\ Charles R Greathouse IV, Feb 09 2017

Extensions

Edited by Robert G. Wilson v, Aug 27 2002
a(16) corrected by T. D. Noe, Feb 08 2008

A097328 Denominator of 1 + 1/5 + 1/9 +...+ 1/(4n+1).

Original entry on oeis.org

1, 5, 45, 585, 9945, 69615, 348075, 10094175, 111035925, 4108329225, 168441498225, 56147166075, 393030162525, 20830598613825, 395781373662675, 24142663793423175, 24142663793423175, 555281267248733025
Offset: 0

Views

Author

T. D. Noe, Aug 04 2004

Keywords

Comments

The first 11 terms are the same as the least common multiples in A051539.

Crossrefs

Cf. A016813 (4n+1), A075136 (numerators), A051539 (lcm of 5, 9, 13, ..., 4n+1).

Programs

  • Mathematica
    Table[Denominator[Total[1/Range[1, 4n+1, 4]]], {n, 0, 20}]
Showing 1-5 of 5 results.

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