A045636 Numbers of the form p^2 + q^2, with p and q primes.

8, 13, 18, 29, 34, 50, 53, 58, 74, 98, 125, 130, 146, 170, 173, 178, 194, 218, 242, 290, 293, 298, 314, 338, 365, 370, 386, 410, 458, 482, 530, 533, 538, 554, 578, 650, 698, 722, 818, 845, 850, 866, 890, 962, 965, 970, 986, 1010, 1058, 1082, 1130, 1202, 1250

Offset: 1

Author

Felice Russo

Keywords

• nonn
• nice

Comments

A045698(a(n)) > 0. - Reinhard Zumkeller, Jul 29 2012

All terms greater than 8 are of the form 8k+2 or 8k+5 (A047617). - Giuseppe Melfi, Oct 06 2022

Examples

			18 belongs to the sequence because it can be written as 3^2 + 3^2.
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• Index entries for sequences related to sums of squares

Crossrefs

A214723 is a subsequence. Complement: A214879.

Cf. A214511 (least number having n orderless representations as p^2 + q^2).

Cf. A047617.

Programs

Haskell

import Data.List (findIndices)
a045636 n = a045636_list !! (n-1)
a045636_list = findIndices (> 0) a045698_list
-- Reinhard Zumkeller, Jul 29 2012

Mathematica

q=13; imax=Prime[q]^2; Select[Union[Flatten[Table[Prime[x]^2+Prime[y]^2, {x,q}, {y,x}]]], #<=imax&] (* Vladimir Joseph Stephan Orlovsky, Apr 20 2011 *)
With[{nn=60},Take[Union[Total/@(Tuples[Prime[Range[nn]],2]^2)],nn]] (* Harvey P. Dale, Jan 04 2014 *)

PARI

list(lim)=my(p1=vector(primepi(sqrt(lim-4)),i,prime(i)^2), t, p2=List()); for(i=1,#p1, for(j=i,#p1, t=p1[i]+p1[j];if(t>lim, break, listput(p2,t)))); vecsort(Vec(p2),,8) \\ Charles R Greathouse IV, Jun 21 2012

Python

from sympy import primerange
def aupto(limit):
    primes = list(primerange(2, int((limit-4)**.5)+2))
    nums = [p*p + q*q for i, p in enumerate(primes) for q in primes[i:]]
    return sorted(set(k for k in nums if k <= limit))
print(aupto(1251)) # Michael S. Branicky, Aug 13 2021