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%I #28 Jul 23 2021 08:59:54
%S 1,3,17,137,227,977,1187,1493,5777,5993
%N Odd numbers not of the form p + 2*k^2, k>0, p prime.
%C This sequence is probably finite.
%C 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
%C The next term, if it exists, is larger than 2 * 10^13. - _Benjamin Chaffin_, Mar 28 2008
%C Terms greater than 3 are of the form 6*n-1. - _Dan Graham_, Mar 03 2015
%D David Wells, Penguin Dictionary of Curious and Interesting Numbers, Revised Edition, Penguin Books, London, 1997, page 76.
%H Laurent Hodges, <a href="https://www.jstor.org/stable/2690477">A lesser-known Goldbach conjecture</a>, Math. Mag., 66 (1993), 45-47.
%H Mark VandeWettering, <a href="https://brainwagon.org/2006/09/05/toying-with-a-lesser-known-goldbach-conjecture/">Toying with a lesser known Goldbach Conjecture</a>, 2006.
%t 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 } ]
%o (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
%Y Cf. A042978.
%K nonn,hard,more
%O 1,2
%A _Robert G. Wilson v_, Mar 14 2001
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