A264499 Numbers n that are the product of four distinct odd primes and x^2 + y^2 = n has integer solutions.

32045, 40885, 45305, 58565, 67405, 69745, 77285, 80665, 91205, 98345, 98605, 99905, 101065, 107185, 111605, 114985, 120445, 124865, 127465, 128945, 130645, 137605, 141245, 146705, 150365, 151385, 162565, 164645, 166685, 167765, 173485, 175565, 179945, 182845

Offset

1,1

Comments

The four primes are of the form 4*k + 1.

Links

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from Colin Barker)

Example

32045 is in the sequence because x^2 + y^2 = 32045 = 5*13*17*29 has solutions (x,y) = (2,179), (19,178), (46,173), (67,166), (74,163), (86,157), (109,142) and (122,131).

Prog

(PARI)
dop(d, nmax) = {
  my(L=List(), v=vector(d, m, 1)~, f);
  for(n=1, nmax,
    f=factorint(n);
    if(#f~==d && f[1, 1]>2 && f[, 2]==v && f[, 1]%4==v, listput(L, n))
  );
  Vec(L)
}
dop(4, 200000)
(Python)
from math import isqrt
from sympy import primerange, integer_nthroot
from oeis_sequences.OEISsequences import bisection
def A264499(n):
    def g(x): return sum(1 for p in primerange(5, x+1) if p&3==1)
    def h(x, y, i): return enumerate((p for p in primerange(x, y) if p&3==1), i)
    def f(x): return int(n+x-sum(g(x//(k*m*r))-c for a, k in h(5, integer_nthroot(x, 4)[0]+1, 1) for b, m in h(k+1, integer_nthroot(x//k, 3)[0]+1, a+1) for c, r in h(m+1, isqrt(x//(k*m))+1, b+1)))
    return bisection(f, n, n) # Chai Wah Wu, Dec 21 2025

Crossrefs

Cf. A131574, A264498.

Sequence in context: A217368 A359343 A097282 * A249230 A250897 A366512

Adjacent sequences: A264496 A264497 A264498 * A264500 A264501 A264502

Keyword

nonn

Author

Colin Barker, Nov 15 2015

Status

approved