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 1-3 of 3 results.

A002091 From a Goldbach conjecture: the location of records in A185091.

Original entry on oeis.org

3, 9, 19, 21, 55, 115, 193, 323, 611, 1081, 1571, 10771, 13067, 16321, 44881, 57887, 93167, 189947, 404939, 442307, 1746551, 3383593, 3544391, 5056787, 7480667, 25619213, 87170987, 404940757, 526805663, 707095391, 1009465507, 1048720723, 5315914139
Offset: 1

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Author

N. J. A. Sloane

Keywords

Comments

A stronger version of the second Goldbach conjecture (every odd number can be expressed as the sum of 3 primes) states that every odd number k > 5 can be written as k = 2*p + q, p, q prime. The conjecture was posed by E. Lemoine and later by H. Levy. The article by B. H. Mayoh assumes q {1,prime}. For the representations of k minimizing q, the sequence gives the value of k at which a larger q than for all representations of j < k is required. The new record value of q is given in A002092. The corresponding sequences for q prime and q=1 excluded are A194828 and A194829. - Hugo Pfoertner, Sep 03 2011
k is in this list when (k+1)/2 is the index of a record in A185091.
Checked up to k=10^13. a(50) is > 10^13. - Hugo Pfoertner, Sep 25 2011

Examples

			a(3)=19, because it is the first number for which q=5 is required. 3=2*1+1, 5=2*2+1, 7=2*3+1, 9=2*3+3, 11=2*5+1, 13=2*5+3, 15=2*7+1, 17=2*7+3, 19=2*7+5.

References

  • Brian H. Mayoh, On the second Goldbach conjecture, Nordisk Tidskr. Informations-Behandling 6, 1966, 48-50.
  • Emile Lemoine, L'intermédiaire des mathématiciens, 1 (1894), 179; ibid 3 (1896), 151.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A002092 [values of q], A194828, A194829 [equivalent with q=1 excluded].
Cf. A185091.

Extensions

a(19)-a(32) from Hugo Pfoertner, Sep 03 2011
a(33) from Jason Kimberley, a(34)-a(40) from Hugo Pfoertner, Sep 09 2011
a(41)-a(49) from Hugo Pfoertner, Sep 25 2011

A002092 From a Goldbach conjecture: records in A185091.

Original entry on oeis.org

1, 3, 5, 7, 17, 29, 47, 61, 73, 83, 277, 317, 349, 419, 503, 601, 709, 829, 877, 1129, 1237, 1367, 1429, 1669, 1801, 2467, 2833, 2879, 3001, 3037, 3329, 3821, 4861, 5003, 5281, 5821, 5897, 6301, 6329, 6421, 6481, 6841, 7069, 7121, 7309, 7873, 8017, 8597, 8821
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

See A002091. The sequence gives the record values of q in the representations minimizing q of 2*k+1 = 2*p+q, p prime, q {1,prime}.
Checked up to 2*k = 10^13.

References

  • Brian H. Mayoh, On the second Goldbach conjecture. Nordisk Tidskr. Informations-Behandling 6, 1966, pp. 48-50.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A002372, A002373, A002374, A002375, A045917, A006307, A002091, A185091, A194828, A194829.

Formula

a(n) = A185091((A002091(n)+1)/2).

Extensions

Comment added, a(19)-a(32) from Hugo Pfoertner, Sep 03 2011
a(33) from Jason Kimberley, a(34)-a(40) from Hugo Pfoertner, Sep 09 2011
a(41)-a(49) from Hugo Pfoertner, Sep 25 2011

A195352 Smallest prime p such that 2*n+1 = 2*p + q for some odd prime q.

Original entry on oeis.org

2, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 3, 7, 2, 2, 3, 5, 2, 3, 2, 2, 3, 2, 3, 7, 2, 3, 7, 2, 2, 3, 5, 2, 3, 2, 2, 3, 5, 2, 3, 2, 3, 19, 2, 3, 7, 5, 2, 3, 2, 2, 3, 2, 2, 3, 2, 3, 7, 5, 11, 7, 11, 2, 3, 2, 3, 13, 2, 2, 3, 5, 5, 7, 2, 2, 3, 5, 2, 3, 7, 2, 3, 2, 3, 13
Offset: 3

Views

Author

Hugo Pfoertner, Sep 16 2011

Keywords

Comments

Related to Lemoine's conjecture, which states that all odd integers > 5 can be represented as 2*p+q, p, q primes.

Examples

			a(3)=2 because 2*3+1=7=2*2+3; a(4)=2: 2*3+1=9=2*2+5; a(5)=2: 11=2*2+7; a(6)=3: 13=2*3+7.

References

Links

Crossrefs

Cf. A002091, A103506 (smallest q), A046927, A195353 (positions of records), A195354 (records), A103153 (prime p=2 excluded), A185091.

Programs

  • Mathematica
    spp[n_]:=Module[{p=2},While[CompositeQ[(2n+1)-2p],p=NextPrime[p]];p]; Array[ spp,90,3] (* Harvey P. Dale, Jun 02 2022 *)
Showing 1-3 of 3 results.

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

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