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

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

Author

N. J. A. Sloane

Keywords

• nonn

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

• 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

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