A060003 Odd numbers not of the form p + 2*k^2, k>0, p prime.

1, 3, 17, 137, 227, 977, 1187, 1493, 5777, 5993

Offset: 1

Author

Robert G. Wilson v, Mar 14 2001

Keywords

• nonn
• hard
• more

Comments

This sequence is probably finite.

Goldbach conjectured that all odd composites are sum of a prime and twice a square. a(9) = 5777 and a(10) = 5993 are the only known exceptions. Elements a(2) .. a(8) are the odd Stern primes (cf. A042978). The next element of the sequence, if it exists, is larger than 10^9. - M. F. Hasler, Nov 16 2007

The next term, if it exists, is larger than 2 * 10^13. - Benjamin Chaffin, Mar 28 2008

Terms greater than 3 are of the form 6*n-1. - Dan Graham, Mar 03 2015

References

• David Wells, Penguin Dictionary of Curious and Interesting Numbers, Revised Edition, Penguin Books, London, 1997, page 76.

Links

• Laurent Hodges, A lesser-known Goldbach conjecture, Math. Mag., 66 (1993), 45-47.
• Mark VandeWettering, Toying with a lesser known Goldbach Conjecture, 2006.

Crossrefs

Cf. A042978.

Programs

Mathematica

Do[ k = 1; While[ n - 2*k^2 > 1 && !PrimeQ[ n - 2*k^2 ], k++ ]; If[ n - 2*k^2 < 0, Print[n] ], { n, 5, 10^8 } ]

PARI

forstep( n=1,2^30,2, for(s=1,sqrtint(n\2), if(isprime(n-2*s^2),next(2)));print(n)) \\ M. F. Hasler, Nov 16 2007