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 A111925 Numbers of the form a^2 + b^4, with a,b > 0. 29
 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 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 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 November 28 06:42 EST 2021. Contains 349401 sequences. (Running on oeis4.)