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

2, 5, 10, 17, 20, 25, 26, 32, 37, 41, 50, 52, 65, 80, 82, 85, 90, 97, 101, 106, 116, 117, 122, 130, 137, 145, 160, 162, 170, 181, 185, 197, 202, 212, 225, 226, 241, 250, 257, 260, 265, 272, 277, 281, 290, 292, 305, 306, 320, 325, 337, 340, 356, 362, 370, 377

Offset: 1

Author

Stefan Steinerberger, Nov 25 2005

Keywords

• nonn

Comments

Subsequence of A000404.

Although there are squares, cubes, fifth powers, ... in this sequence, there are no fourth powers. - Altug Alkan, Apr 09 2016

Also, numbers z such that z^5 = x^2 + y^4 for x, y >= 1. - M. F. Hasler, Apr 16 2018

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

Examples

			25 = 3^2 + 2^4, so 25 is an element of the sequence.
		

Links

• R. J. Mathar, Table of n, a(n) for n = 1..1000
• J. Friedlander and H. Iwaniec, The polynomial x^2 + y^4 captures its primes, arXiv:math/9811185 [math.NT], 1998; Ann. of Math. 148 (1998), 945-1040.
• Wikipedia, Friedlander-Iwaniec theorem

Crossrefs

Cf. A055394, A022549; complement of A111909; subsequence of A000404.

Cf. A028916 (subsequence of primes).

Programs

Maple

isA111925 := proc(n)
    local a,b ;
    for a from 1 do
        if a^4 >= n then
            return false;
        end if;
        b := n-a^4 ;
        if issqr(b) then
            return true;
        end if;
    end do:
end proc:
A111925 := proc(n)
    option remember;
    if n = 1 then
        2;
    else
        for a from procname(n-1)+1 do
            if isA111925(a) then
                return a;
            end if;
        end do:
    end if;
end proc: # R. J. Mathar, Apr 22 2013

Mathematica

With[{nn=60},Take[Union[First[#]^2+Last[#]^4&/@Tuples[Range[nn],2]],nn]] (* Harvey P. Dale, Jul 09 2014 *)

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

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