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 1-4 of 4 results.

A124837 Numerators of third-order harmonic numbers (defined by Conway and Guy, 1996).

Original entry on oeis.org

1, 7, 47, 57, 459, 341, 3349, 3601, 42131, 44441, 605453, 631193, 655217, 1355479, 23763863, 24444543, 476698557, 162779395, 166474515, 34000335, 265842403, 812400067, 20666950267, 21010170067, 192066102203, 194934439103
Offset: 1

Views

Author

Jonathan Vos Post, Nov 10 2006

Keywords

Comments

Denominators are A124838. All fractions reduced. Thanks to Jonathan Sondow for verifying these calculations. He suggests that the equivalent definition in terms of first order harmonic numbers may be computationally simpler. We are happy with the description of A027612 Numerator of 1/n + 2/(n-1) + 3/(n-2) + ... + (n-1)/2 + n, but baffled by the description of A027611.
From Alexander Adamchuk, Nov 11 2006: (Start)
a(n) is the numerator of H(n, (3)) = Sum_{m=1..n} Sum_{k=1..m} HarmonicNumber(k).
Denominators are listed in A124838.
p divides a(p-5) for prime p > 5.
Primes are listed in A129880.
Numbers k such that a(k) is prime are listed in A129881. (End)

Examples

			a(1) = 1 = numerator of 1/1.
a(2) = 7 = numerator of 1/1 + 5/2 = 7/2.
a(3) = 47 = numerator of 7/2 + 13/3 = 47/6.
a(4) = 57 = numerator of 47/6 + 77/12 = 57/4.
a(5) = 549 = numerator of 57/4 + 87/10 = 549/20.
a(6) = 341 = numerator of 549/20 + 223/20 = 341/10
a(7) = 3349 = numerator of 341/10 + 481/35 = 3349/70.
a(8) = 88327 = numerator of 3349/70 + 4609/280 = 88327/1260.
a(9) = 3844 = numerator of 88327/1260 + 4861/252 = 3844/45.
a(10) = 54251 = numerator of 3844/45 + 55991/2520 = 54251/504, or, untelescoping:
a(10) = 54251 = numerator of 1/1 + 5/2 + 13/3 + 77/12 + 87/10 + 223/20 + 481/35 + 4609/252 + 4861/252 + 55991/2520 = 54251/504.

References

  • J. H. Conway and R. K. Guy, The Book of Numbers, New York: Springer-Verlag, pp. 143 and 258-259, 1996.

Links

Crossrefs

Cf. A027611, A027612, A124838.
Cf. A001008, A002805, A067657, A056903, A124878, A124879, A124837, A129880, A129881.

Programs

  • Haskell
    a124837 n = a213998 (n + 2) (n - 1) -- Reinhard Zumkeller, Jul 03 2012
  • Mathematica
    Table[Numerator[(n+2)!/2!/n!*Sum[1/k,{k,3,n+2}]],{n,1,30}] (* Alexander Adamchuk, Nov 11 2006 *)

Formula

A124837(n)/A124838(n) = Sum{i=1..n} A027612(n)/A027611(n+1).
From Alexander Adamchuk, Nov 11 2006: (Start)
a(n) = numerator(Sum_{m=1..n} Sum_{l=1..m} Sum_{k=1..l} 1/k).
a(n) = numerator(((n+2)!/(2!*n!)) * Sum_{k=3..n+2} 1/k).
a(n) = numerator(((n+2)*(n+1)/2) * Sum_{k=3..n+2} 1/k). (End)
a(n) = numerator(Sum_{k=0..n-1} (-1)^k*binomial(-3,k)/(n-k)). - Gary Detlefs, Jul 18 2011
a(n) = A213998(n+2,n-1). - Reinhard Zumkeller, Jul 03 2012

Extensions

Corrected and extended by Alexander Adamchuk, Nov 11 2006

A124878 Primes in A027612.

Original entry on oeis.org

5, 13, 223, 4861, 197698279, 25472027467, 6975593267347, 218572480850557, 1592457339642613, 2955634782407818711841368777079578319, 2950127241932882597818337002939124083061, 232242878286351670588710938679161483012314573381293769
Offset: 1

Views

Author

Alexander Adamchuk, Nov 11 2006

Keywords

Examples

			A027612(n) begins {1, 5, 13, 77, 87, 223, 481, 4609, 4861, ...}.
Thus a(1) = 5, a(2) = 13, a(3) = 223, a(4) = 4861.

Links

Crossrefs

A027612(n) are the numerators of second order harmonic numbers H(n, (2)).
Corresponding numbers n such that A027612(n) is prime are listed in A124879.
Cf. A001008, A002805, A067657, A056903, A027612, A124879, A124837, A124880, A124881.

Programs

  • Mathematica
    s=1;Do[s=s+1/(n+1);f=Numerator[(n+1)*(s-1)]; If[PrimeQ[f],Print[{n,f}]],{n,1,1942}]
  • PARI
    lista(nn) = {for (n=1, nn, if (isprime(p=numerator(sum(k=1, n, k/(n-k+1)))), print1(p, ", ")););} \\ Michel Marcus, Jul 14 2018

Formula

a(n) = A027612(A124879(n)).

Extensions

a(12) from, and crossrefs edited by Michel Marcus, Jul 14 2018

A124880 Primes in A124837.

Original entry on oeis.org

7, 47, 42131, 23763863, 192066102203, 5733412167187, 34745876421709, 185813891783454008069, 171312804637561107990389, 29207630124216024960052176833, 6300447575454970515437116064749
Offset: 1

Views

Author

Alexander Adamchuk, Nov 11 2006

Keywords

Examples

			A124837(n) begins {1, 7, 47, 57, 459, 341, 3349, 3601, 42131, 44441, ...}.
Thus a(1) = 7, a(2) = 47, a(3) = 42131.

Links

Crossrefs

A124837 are the numerators of third-order harmonic numbers H(n, (3)).
Corresponding numbers n such that A124837(n) is prime are listed in A124881.
Cf. A001008, A002805, A067657, A056903, A027612, A124878, A124879, A124837, A124881.

Programs

  • Mathematica
    s=3/2;Do[s=s+1/n;f=Numerator[n*(n-1)/2*(s-3/2)]; If[PrimeQ[f],Print[{n-2,f}]],{n,3,125}]

Extensions

Crossrefs edited by Michel Marcus, Jul 14 2018

A124881 Numbers k such that A124837(k) is prime.

Original entry on oeis.org

2, 3, 9, 15, 25, 27, 33, 45, 55, 67, 70, 93, 94, 97, 112, 113, 125, 137, 189, 193, 212, 232, 262, 273, 281, 381, 453, 528, 670, 677, 742, 743, 827, 996, 1257, 1349, 1402, 1645, 1683, 2110, 2217, 2408, 2480, 2623, 3208, 3517, 3637, 3665, 4571, 4730
Offset: 1

Views

Author

Alexander Adamchuk, Nov 11 2006

Keywords

Examples

			A124837(n) begins {1, 7, 47, 57, 459, 341, 3349, 3601, 42131, 44441, ...}.
Thus a(1) = 2, a(2) = 3, a(3) = 9.

Links

Crossrefs

A124837 are the numerators of third-order harmonic numbers H(n, (3)).
Corresponding primes in A124837 are listed in A124880.
Cf. A001008, A002805, A067657, A056903, A027612, A124878, A124879, A124837, A124880.

Programs

  • Mathematica
    s=3/2;Do[s=s+1/n;f=Numerator[n*(n-1)/2*(s-3/2)]; If[PrimeQ[f],Print[{n-2,f}]],{n,3,1000}]

Extensions

More terms from Stefan Steinerberger, May 09 2007
Crossrefs edited by Michel Marcus, Jul 14 2018
Showing 1-4 of 4 results.

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