A214879 -id:A214879 - cp's OEIS frontend

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

Felice Russo

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

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

A214723 is a subsequence. Complement: A214879.
Cf. A214511 (least number having n orderless representations as p^2 + q^2).
Cf. A047617.

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

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 *)

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

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

A045698 Number of ways n can be written as the sum of two squares of primes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0

Offset: 0

Felice Russo

a(A214879(n)) = 0; a(A045636(n)) > 0; a(A214723(n)) = 1; a(A214511(n)) = n and a(m) < n for m < A214511(n). - Reinhard Zumkeller, Jul 29 2012

The smallest value of n such that a(n) = 2 is 338. (This helps distinguish it from the characteristic function of A045636.) - Wesley Ivan Hurt, Jun 13 2013

			For example, a(29) = 1 because 29 = 2^2 + 5^2. a(3) = 0 because there is no way to write 3 as sum of two squares of primes.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for sequences related to sums of squares

Cf. A001248, A010051, A037213.

a045698 n = length $ filter (\x -> x > 0 && a010051' x == 1) $
map (a037213 . (n -)) $
takeWhile (<= div n 2) a001248_list
-- Reinhard Zumkeller, Jul 29 2012

a(n)=my(s=0,q);forprime(p=2,sqrtint(n\2),if(issquare(n-p^2,&q)&&isprime(q),s++));s \\ Charles R Greathouse IV, Jun 04 2014

More terms from Erich Friedman

A283248 Numbers that cannot be represented as a sum of one prime and two primes squared.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 14, 17, 22, 28, 33, 38, 43, 62, 68, 78, 83, 98, 104, 134, 146, 158, 174, 179, 182, 188, 200, 218, 230, 248, 260, 266, 272, 278, 302, 308, 314, 328, 332, 338, 356, 374, 398, 404, 416, 428, 440, 458, 464, 482, 488, 494, 506, 518, 524, 530

Offset: 1

Michel Marcus, Mar 05 2017

nonn

			10 is the first missing number since it is 2 + 2^2 + 2^2.

Michel Marcus, Table of n, a(n) for n = 1..1000
Hongze Li, Sums of one prime and two prime squares, Acta Arithmetica (2008), Volume: 134, Issue: 1, page 1-9.
Lilu Zhao, The additive problem with one prime and two squares of primes, Journal of Number Theory, Volume 135, February 2014, Pages 8-27.

Cf. A214879.

isok2(n)=forprime(q=2,sqrtint(n),if(isprimepower(n-q^2)==2, return(0))); 1;
isok(n) = forprime(p=2, n-1, if (!isok2(n-p), return (0));); 1;

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

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A045698 Number of ways n can be written as the sum of two squares of primes.

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A283248 Numbers that cannot be represented as a sum of one prime and two primes squared.

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