cp's OEIS Frontend

A111925 Numbers of the form a^2 + b^4, with a,b > 0.

%I A111925 #32 Aug 04 2021 02:17:39
%S A111925 2,5,10,17,20,25,26,32,37,41,50,52,65,80,82,85,90,97,101,106,116,117,
%T A111925 122,130,137,145,160,162,170,181,185,197,202,212,225,226,241,250,257,
%U A111925 260,265,272,277,281,290,292,305,306,320,325,337,340,356,362,370,377
%N A111925 Numbers of the form a^2 + b^4, with a,b > 0.
%C A111925 Subsequence of A000404.
%C A111925 Although there are squares, cubes, fifth powers, ... in this sequence, there are no fourth powers. - _Altug Alkan_, Apr 09 2016
%C A111925 Also, numbers z such that z^5 = x^2 + y^4 for x, y >= 1. - _M. F. Hasler_, Apr 16 2018
%C A111925 The Friedlander-Iwaniec theorem states that there are infinitely many prime numbers in this sequence. These primes are in A028916. - _Bernard Schott_, Mar 09 2019
%H A111925 R. J. Mathar, <a href="/A111925/b111925.txt">Table of n, a(n) for n = 1..1000</a>
%H A111925 J. Friedlander and H. Iwaniec, <a href="https://arxiv.org/abs/math/9811185">The polynomial x^2 + y^4 captures its primes</a>, arXiv:math/9811185 [math.NT], 1998; Ann. of Math. 148 (1998), 945-1040.
%H A111925 Wikipedia, <a href="https://en.wikipedia.org/wiki/Friedlander%E2%80%93Iwaniec_theorem">Friedlander-Iwaniec theorem</a>
%e A111925 25 = 3^2 + 2^4, so 25 is an element of the sequence.
%p A111925 isA111925 := proc(n)
%p A111925     local a,b ;
%p A111925     for a from 1 do
%p A111925         if a^4 >= n then
%p A111925             return false;
%p A111925         end if;
%p A111925         b := n-a^4 ;
%p A111925         if issqr(b) then
%p A111925             return true;
%p A111925         end if;
%p A111925     end do:
%p A111925 end proc:
%p A111925 A111925 := proc(n)
%p A111925     option remember;
%p A111925     if n = 1 then
%p A111925         2;
%p A111925     else
%p A111925         for a from procname(n-1)+1 do
%p A111925             if isA111925(a) then
%p A111925                 return a;
%p A111925             end if;
%p A111925         end do:
%p A111925     end if;
%p A111925 end proc: # _R. J. Mathar_, Apr 22 2013
%t A111925 With[{nn=60},Take[Union[First[#]^2+Last[#]^4&/@Tuples[Range[nn],2]],nn]] (* _Harvey P. Dale_, Jul 09 2014 *)
%o A111925 (PARI) list(lim)=my(v=List(),t); lim\=1; for(b=1,sqrtnint(lim-1,4), t=b^4; for(a=1,sqrtint(lim-t), listput(v,t+a^2))); Set(v) \\ _Charles R Greathouse IV_, Jun 07 2016
%o A111925 (PARI) is(n)=for(b=1,sqrtnint(n-1,4), if(issquare(n-b^4), return(1))); 0 \\ _Charles R Greathouse IV_, Jun 07 2016
%Y A111925 Cf. A055394, A022549; complement of A111909; subsequence of A000404.
%Y A111925 Cf. A028916 (subsequence of primes).
%K A111925 nonn
%O A111925 1,1
%A A111925 _Stefan Steinerberger_, Nov 25 2005