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

%I

%S 13,29,41,53,61,73,89,97,109,113,137,149,157,173,181,193,229,233,241,

%T 269,277,281,293,313,317,337,349,353,373,389,397,409,421,433,449,457,

%U 461,509,521,541,557,569,593,601,613,617,641,653,661,673,701,709,733,757,761,769

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

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

%C Same as A002144 with A002496 removed.

%D 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.

%F a(n) ~ 2n log n. - _Charles R Greathouse IV_, Jun 10 2017

%t Complement[Select[4*Range[400]+1, PrimeQ], Select[Range[40]^2+1, PrimeQ]] - _T. D. Noe_, Jun 27 2008

%t Select[Prime[Range[200]],IntegerQ[(#-1)/4]&&!IntegerQ[Sqrt[#-1]]&] (* _Harvey P. Dale_, Jan 04 2015 *)

%o (PARI) forprime(p=3,1000,if(p%4==1&&!issquare((p-1)/4),print1(p,", "))) \\ _Joerg Arndt_, Jul 01 2012

%o (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

%Y Cf. A002144, A002145, A002496.

%K nonn

%O 1,1

%A _A.K. Devaraj_, Jun 23 2008

%E Corrected and extended by T. D. Noe and _N. J. A. Sloane_, Jun 27 2008

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Last modified January 19 21:35 EST 2018. Contains 297938 sequences.