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A111925 Numbers of the form a^2 + b^4, with a,b > 0. 24
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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

LINKS

R. J. Mathar, Table of n, a(n) for n = 1..1000

Wikipedia, Friedlander-Iwaniec theorem

EXAMPLE

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

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 *)

PROG

(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

CROSSREFS

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

Cf. A028916 (subsequence of primes).

Sequence in context: A067112 A101306 A051351 * A238804 A030723 A077166

Adjacent sequences:  A111922 A111923 A111924 * A111926 A111927 A111928

KEYWORD

nonn

AUTHOR

Stefan Steinerberger, Nov 25 2005

STATUS

approved

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Last modified December 8 04:31 EST 2019. Contains 329850 sequences. (Running on oeis4.)