

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
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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 FriedlanderIwaniec 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, FriedlanderIwaniec 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 := na^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(n1)+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(lim1, 4), t=b^4; for(a=1, sqrtint(limt), listput(v, t+a^2))); Set(v) \\ Charles R Greathouse IV, Jun 07 2016
(PARI) is(n)=for(b=1, sqrtnint(n1, 4), if(issquare(nb^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



