A002091 From a Goldbach conjecture: the location of records in A185091. (Formerly M2773 N1116)

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

1,1

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

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

Hugo Pfoertner, Table of n, a(n) for n = 1..49

Brian H. Mayoh, On the second Goldbach conjecture, BIT Numerical Mathematics 6 (1966) 1, 48-50

Wikipedia, Lemoine's conjecture

Index entries for sequences related to Goldbach conjecture

Example

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.

Crossrefs

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

Cf. A185091.

Sequence in context: A097267 A043097 A084670 * A056259 A376220 A056682

Adjacent sequences: A002088 A002089 A002090 * A002092 A002093 A002094

Keyword

nonn

Author

N. J. A. Sloane

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

Status

approved