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
KEYWORD
nonn
AUTHOR
Stefan Steinerberger, Nov 25 2005
STATUS
approved