A141293 Primes p of the form 4*k+1 which are not of the form r^2 + 1.

13, 29, 41, 53, 61, 73, 89, 97, 109, 113, 137, 149, 157, 173, 181, 193, 229, 233, 241, 269, 277, 281, 293, 313, 317, 337, 349, 353, 373, 389, 397, 409, 421, 433, 449, 457, 461, 509, 521, 541, 557, 569, 593, 601, 613, 617, 641, 653, 661, 673, 701, 709, 733, 757, 761, 769

Offset: 1

Author

A.K. Devaraj, Jun 23 2008

Keywords

• nonn

Comments

Equivalently, prime factors of numbers of the form x^2 + 1 which themselves are not of this form.

Same as A002144 with A002496 removed.

References

• A. K. Devaraj, "Euler's Generalization of Fermat's Theorem-A Further Generalization", in ISSN #1550-3747, Proceedings of Hawaii Intl Conference on Statistics, Mathematics & Related Fields, 2004.

Crossrefs

Cf. A002144, A002145, A002496.

Programs

Mathematica

Complement[Select[4*Range[400]+1, PrimeQ], Select[Range[40]^2+1, PrimeQ]] (* T. D. Noe, Jun 27 2008 *)
Select[Prime[Range[200]],IntegerQ[(#-1)/4]&&!IntegerQ[Sqrt[#-1]]&] (* Harvey P. Dale, Jan 04 2015 *)

PARI

forprime(p=3,1000,if(p%4==1&&!issquare((p-1)/4),print1(p,", "))) \\ Joerg Arndt, Jul 01 2012

PARI

list(lim)=my(v=List()); forprime(p=2,lim, if(p%4==1, listput(v,p))); v=setminus(Set(v), vector(sqrtint(lim\4),i,4*i^2+1)) \\ Charles R Greathouse IV, Jun 10 2017

Formula

a(n) ~ 2n log n. - Charles R Greathouse IV, Jun 10 2017

Extensions

Corrected and extended by T. D. Noe and N. J. A. Sloane, Jun 27 2008