A042978 Stern primes: primes not of the form p + 2b^2 for p prime and b > 0.

2, 3, 17, 137, 227, 977, 1187, 1493

Offset: 1

Author

Jud McCranie

Keywords

• nonn
• more

Comments

No others < 1299709. Are there any others? Related to a conjecture of Goldbach.

The next element of the sequence, if it exists, is larger than 10^9 ; see A060003. - M. F. Hasler, Nov 16 2007

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

Does not equal A000040(k) + A001105(j) for all k & j >0. - Robert G. Wilson v, Sep 07 2012

References

• Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966. See p. 226.
• J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 137, p. 46, Ellipses, Paris 2008.
• L. E. Dickson, History of the theory of Numbers, vol. 1, page 424.

Links

• Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
• L. Hodges, A lesser-known Goldbach conjecture, Math. Mag., 66 (1993), 45-47.
• Mark VandeWettering, Toying with a lesser known Goldbach Conjecture
• Index entries for sequences related to Goldbach conjecture

Crossrefs

Apart from the first term, a subsequence of A060003.

Programs

Maple

N:= 10^6: # to check primes up to N
P:= select(isprime, {2,seq(i,i=3..N,2)}):
S:= {seq(2*b^2,b=1..floor(sqrt(N/2)))}:
P minus {seq(seq(p+s,p=P),s=S)}; # Robert Israel, Jan 19 2016

Mathematica

fQ[n_] := Block[{k = Floor[ Sqrt[ n/2]]}, While[k > 0 && !PrimeQ[n - 2*k^2], k--]; k == 0]; Select[ Prime[Range[238]], fQ] (* Robert G. Wilson v, Sep 07 2012 *)

PARI

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

PARI

forprime(p=2,4e9,forstep(k=sqrt(p\2),1,-1,if(isprime(p-2*k^2),next(2)));print1(p", ")) \\ Charles R Greathouse IV, Aug 04 2011