A214514 Numbers of the form p^2 + q^2 + r^2, where p, q, and r are primes.

12, 17, 22, 27, 33, 38, 43, 54, 57, 59, 62, 67, 75, 78, 83, 99, 102, 107, 123, 129, 134, 139, 147, 150, 155, 171, 174, 177, 179, 182, 187, 195, 198, 203, 219, 222, 227, 243, 246, 251, 267, 291, 294, 297, 299, 302, 307, 315, 318, 323, 339, 342, 347, 363, 369

Offset: 1

Author

T. D. Noe, Jul 29 2012

Keywords

• nonn

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A045636 (two primes), A214515 (four primes).

Programs

Mathematica

nn = 10^3; ps = Prime[Range[PrimePi[Sqrt[nn]]]]; t = Flatten[Table[ps[[i]]^2 + ps[[j]]^2 + ps[[k]]^2, {i, Length[ps]}, {j, i, Length[ps]}, {k, j, Length[ps]}]]; t = Select[t, # <= nn &]; Union[t]

Python

from sympy import primerange as primes
from itertools import takewhile, combinations_with_replacement as mc
def aupto(N):
    psqs = list(takewhile(lambda x: x<=N, (p**2 for p in primes(1, N+1))))
    sum3 = set(sum(c) for c in mc(psqs, 3) if sum(c) <= N)
    return sorted(sum3)
print(aupto(369)) # Michael S. Branicky, Dec 17 2021