A281792 Primes of the form x^2 + p^4 where x > 0 and p is prime.

17, 41, 97, 137, 181, 241, 277, 337, 457, 641, 661, 757, 769, 821, 857, 881, 977, 1109, 1201, 1237, 1301, 1409, 1697, 2017, 2069, 2389, 2417, 2437, 2617, 2657, 2741, 2801, 3041, 3217, 3301, 3329, 3541, 3557, 3697, 3761, 3989, 4001, 4177, 4241, 4337, 4517, 4721, 5557, 5641, 5857, 6101, 6257, 6481, 6577

Offset: 1

Author

Robert Israel, Jan 30 2017

Keywords

• nonn

Comments

Heath-Brown and Li prove an asymptotic formula for the number of terms <= x, in particular showing that the sequence is infinite.

Examples

			17 = 1^2 + 2^4
41 = 5^2 + 2^4
97 = 9^2 + 2^4
137 = 11^2 + 2^4
181 = 10^2 + 3^4
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• D.R. Heath-Brown and X. Li, Prime values of a^2+p^4, arXiv:1504.00531 [math.NT], 2015; Inventiones Mathematicae page 1-59 (Sep 30 2016), doi:10.1007/s00222-016-0694-0

Crossrefs

Subsequence of A028916.

Cf. A199401, A225119.

Programs

Maple

N:= 10000: # to get all terms <= N
A:= select(isprime, {seq(seq(x^2+y^4, x=1..floor(sqrt(N-y^4))),
y=select(isprime, [$1..floor(N^(1/4))]))}):
sort(convert(A,list)); # Robert Israel, Jan 30 2017

Mathematica

nn = 10000;
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 *)

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

Formula

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