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%I #20 Sep 18 2018 09:09:32
%S 17,41,97,137,181,241,277,337,457,641,661,757,769,821,857,881,977,
%T 1109,1201,1237,1301,1409,1697,2017,2069,2389,2417,2437,2617,2657,
%U 2741,2801,3041,3217,3301,3329,3541,3557,3697,3761,3989,4001,4177,4241,4337,4517,4721,5557,5641,5857,6101,6257,6481,6577
%N Primes of the form x^2 + p^4 where x > 0 and p is prime.
%C Heath-Brown and Li prove an asymptotic formula for the number of terms <= x, in particular showing that the sequence is infinite.
%H Robert Israel, <a href="/A281792/b281792.txt">Table of n, a(n) for n = 1..10000</a>
%H D.R. Heath-Brown and X. Li, <a href="https://arxiv.org/abs/1504.00531">Prime values of a^2+p^4</a>, arXiv:1504.00531 [math.NT], 2015; Inventiones Mathematicae page 1-59 (Sep 30 2016), doi:10.1007/s00222-016-0694-0
%F Heath-Brown and Li prove that there are c*x^(3/4)/log^2 x terms up to x, where c = 4*nu*J = 4.79946121442200811438003177..., nu = A199401, and J = A225119. - _Charles R Greathouse IV_, Aug 21 2017
%e 17 = 1^2 + 2^4
%e 41 = 5^2 + 2^4
%e 97 = 9^2 + 2^4
%e 137 = 11^2 + 2^4
%e 181 = 10^2 + 3^4
%p N:= 10000: # to get all terms <= N
%p A:= select(isprime, {seq(seq(x^2+y^4, x=1..floor(sqrt(N-y^4))),
%p y=select(isprime, [$1..floor(N^(1/4))]))}):
%p sort(convert(A,list)); # _Robert Israel_, Jan 30 2017
%t nn = 10000;
%t Select[Table[x^2+y^4, {y, Select[Range[nn^(1/4)], PrimeQ]}, {x, Sqrt[nn-y^4 ]}] // Flatten, PrimeQ] // Union (* _Jean-François Alcover_, Sep 18 2018, after _Robert Israel_ *)
%o (PARI) list(lim)=if(lim<17, return([])); my(v=List(),p4,t); forstep(a=1,sqrtint(-16+lim\=1),2, if(isprime(t=a^2+16), listput(v,t))); forprime(p=3,sqrtnint(lim-4,4), p4=p^4; forstep(a=2,sqrtint(lim-p4),2, if(isprime(t=p4+a^2), listput(v,t)))); Set(v) \\ _Charles R Greathouse IV_, Aug 13 2017
%Y Subsequence of A028916.
%Y Cf. A199401, A225119.
%K nonn
%O 1,1
%A _Robert Israel_, Jan 30 2017
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