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 1-7 of 7 results.

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 *)

A075136 Numerator of the generalized harmonic number H(n,4,1).

Original entry on oeis.org

1, 6, 59, 812, 14389, 104038, 534113, 15837352, 177575597, 6681333014, 278042982799, 93928709068, 665521987201, 35665695484178, 684591747070657, 42155877944972752, 42527303541794647, 986175536059084606
Offset: 1

Views

Author

T. D. Noe, Sep 04 2002

Keywords

Comments

The denominators are in A051539. See A075135 for more details.
Numerators of the partial sums of the divergent series 1/3 + 1/7 + 1/11 + . . 1/(4n-1).

Crossrefs

Cf. A051539, A075135.

Programs

  • Mathematica
    a=4; b=1; maxN=20; s=0; Numerator[Table[s+=1/(a n + b), {n, 0, maxN-1}]]
Numerator[Accumulate[1/Range[1,69,4]]] (* Harvey P. Dale, Dec 15 2014 *)
  • PARI
    sumrecip(n,a,b) = { s=0; default(realprecision,n); forstep(j=b,n,a, s=s+1/j; print1(numerator(s)",") ) }

Formula

Sum 1/a(n) = 1.111939597509272224249... - Cino Hilliard, Dec 21 2003

A051541 Quotients of consecutive values of LCM {1, 5, 9, 13, 17, ..., (4n+1)}.

Original entry on oeis.org

5, 9, 13, 17, 7, 5, 29, 11, 37, 41, 1, 7, 53, 19, 61, 1, 23, 73, 1, 9, 1, 89, 31, 97, 101, 1, 109, 113, 1, 11, 5, 43, 1, 137, 47, 1, 149, 1, 157, 1, 1, 13, 173, 59, 181, 1, 1, 193, 197, 67, 1, 1, 71, 1, 1, 1, 229, 233, 79, 241, 1, 83, 1, 257, 1, 1, 269, 1, 277
Offset: 1

Views

Author

Asher Auel

Keywords

Examples

			a(2) = A051539(2)/A051539(1) = 45/5 = 9.

Crossrefs

Cf. A051539.

Programs

  • Mathematica
    Table[LCM@@Table[4k+1,{k,0,n}]/LCM@@Table[4k+1,{k,0,n-1}],{n,69}] (* Stefano Spezia, Dec 27 2022 *)

Formula

a(n) = A051539(n)/A051539(n-1).

Extensions

More terms from Sean A. Irvine, Sep 17 2021

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}]

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

Views

Author

T. D. Noe, Aug 04 2004

Keywords

Comments

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

Links

Crossrefs

Cf. A004767 (4n+3), A074638 (denominator of 1/3+1/7+/11+...+1/(4n+3)).
Cf. A051539.

Programs

  • Magma
    k:=67; [Lcm([h: h in [3..j by 4]]): j in [3..k by 4]];  // Bruno Berselli, May 03 2011
  • Mathematica
    Table[LCM@@Range[3, 4n+3, 4], {n, 0, 19}]

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}]

A131940 Least common multiple of {1, 7, 13, 19, 25, ..., (6n+1)} (A016921).

Original entry on oeis.org

1, 7, 91, 1729, 43225, 1339975, 49579075, 2131900225, 14923301575, 164156317325, 10013535356825, 670906868907275, 48976201430231075, 3869119912988254925, 65775038520800333725, 65775038520800333725, 6380178736517632371325
Offset: 0

Views

Author

Jonathan Vos Post, Oct 05 2007

Keywords

Comments

This is to 6n+1 (A016921) as A051539 is to 4n+1 (A016813). Because 6*9 + 1 = 49 is divisible by 6*1 + 1 = 7, this sequence differs from A008542. a(n) | A008542(n+1).

Crossrefs

Cf. A008542, A016813, A016921, A051539.

Programs

  • Mathematica
    a = {1}; Do[l = 1; For[j = 1, j < n, j++, l = LCM[l, 6*j + 1]]; AppendTo[a, l], {n, 2, 20}]; a (* Stefan Steinerberger, Oct 07 2007 *)
Join[{1},Table[LCM@@(6*Range[0,n]+1),{n,20}]] (* Harvey P. Dale, Apr 30 2019 *)

Extensions

More terms from Stefan Steinerberger, Oct 07 2007
Showing 1-7 of 7 results.

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