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.

A302192 Denominators of Hurwitz inverse of primes [2,3,5,7,...].

Original entry on oeis.org

2, 4, 1, 8, 2, 4, 4, 16, 2, 4, 1, 8, 2, 2, 8, 32, 2, 4, 1, 8, 2, 4, 4, 16, 1, 4, 4, 8, 2, 8, 16, 64, 1, 4, 4, 8, 2, 4, 8, 16, 2, 4, 4, 8, 1, 4, 16, 32, 2, 4, 4, 8, 1, 4, 2, 16, 1, 4, 4, 8, 2, 16, 16, 128, 1, 4, 4, 8, 2, 2, 8, 16, 1, 4, 1, 8, 1, 8, 16, 32, 1, 4, 1, 8, 2
Offset: 0

Views

Author

N. J. A. Sloane and William F. Keigher, Apr 12 2018

Keywords

Comments

In the ring of Hurwitz sequences all members have offset 0.

Examples

			1/2, -3/4, 1, -5/8, -7/2, 97/4, -403/4, 3795/16, 1683/2, -67403/4, 141662, -5744835/8, -710829/2, 124489961/2, -7187558877/8, ...

Links

  • Xing Gao and William F. Keigher, Interlacing of Hurwitz series, Communications in Algebra, 45:5 (2017), 2163-2185, DOI: 10.1080/00927872.2016.1226885.

Crossrefs

Cf. A302191, A302193.

Programs

  • Maple
    (See A302191 for Maple code)

Formula

E.g.f. for A302191/A302192 is 1 / Sum_{n >= 0} prime(n+1)*x^n/n!.

A302194 Hurwitz inverse of [1 followed by primes], [1,2,3,5,7,...].

Original entry on oeis.org

1, -2, 5, -17, 79, -471, 3391, -28451, 272447, -2933807, 35102403, -462021525, 6634207777, -103200019093, 1728836723813, -31030630439249, 594094812208133, -12085090282079299, 260296103744105623, -5917885334682695549, 141625618336446419151
Offset: 0

Views

Author

N. J. A. Sloane and William F. Keigher, Apr 12 2018

Keywords

Comments

In the ring of Hurwitz sequences all members have offset 0.

References

  • Xing Gao and William F. Keigher, Interlacing of Hurwitz series, Communications in Algebra, 45:5 (2017), 2163-2185, DOI: 10.1080/00927872.2016.1226885

Links

Crossrefs

Cf. A302191, A302192.

Programs

  • Maple
    # first load Maple commands for Hurwitz operations from link
s:=[1, seq(ithprime(n),n=1..64)];
Hinv(s);

Formula

E.g.f. = 1 / (1 + Sum_{n >= 1} prime(n)*x^n/n!).

A302193 a(n) = log_2(A302192(n)).

Original entry on oeis.org

1, 2, 0, 3, 1, 2, 2, 4, 1, 2, 0, 3, 1, 1, 3, 5, 1, 2, 0, 3, 1, 2, 2, 4, 0, 2, 2, 3, 1, 3, 4, 6, 0, 2, 2, 3, 1, 2, 3, 4, 1, 2, 2, 3, 0, 2, 4, 5, 1, 2, 2, 3, 0, 2, 1, 4, 0, 2, 2, 3, 1, 4, 4, 7, 0, 2, 2, 3, 1, 1, 3, 4, 0, 2, 0, 3, 0, 3, 4, 5, 0, 2, 0, 3, 1
Offset: 0

Views

Author

N. J. A. Sloane and William F. Keigher, Apr 12 2018

Keywords

Crossrefs

Cf. A302191, A302192.

A307770 Expansion of e.g.f. 1/(1 - Sum_{k>=1} prime(k)*x^k/k!).

Original entry on oeis.org

1, 2, 11, 89, 957, 12871, 207717, 3910931, 84155053, 2037195551, 54795228241, 1621233039941, 52328310410427, 1829742961027269, 68901415049874055, 2779901582389463177, 119635322278784511015, 5470390958849723994819, 264850557367286330886261, 13535194864326763053170325
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 27 2019

Keywords

Links

Crossrefs

Cf. A000040, A007446, A030017, A302191, A302192, A302194.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, add(
      binomial(n, j)*ithprime(j)*a(n-j), j=1..n))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Jun 24 2021
  • Mathematica
    nmax = 19; CoefficientList[Series[1/(1 - Sum[Prime[k] x^k/k!, {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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