A264498 Numbers n that are the product of three distinct odd primes and x^2 + y^2 = n has integer solutions.

1105, 1885, 2405, 2465, 2665, 3145, 3445, 3485, 3965, 4505, 4745, 5185, 5365, 5785, 5945, 6205, 6305, 6409, 6565, 7085, 7345, 7565, 7585, 7685, 8177, 8245, 8585, 8845, 8905, 9061, 9265, 9605, 9685, 9805, 10205, 10585, 10865, 11245, 11285, 11645, 11713, 11765

Offset

1,1

Comments

The three 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

1105 is in the sequence because x^2 + y^2 = 1105 = 5*13*17 has solutions (x,y) = (4,33), (9,32), (12,31) and (23,24).

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(3, 15000)
(Python)
from math import isqrt
from sympy import primerange, integer_nthroot
from oeis_sequences.OEISsequences import bisection
def A264498(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))-b for a, k in h(5, integer_nthroot(x, 3)[0]+1, 1) for b, m in h(k+1, isqrt(x//k)+1, a+1)))
    return bisection(f, n, n) # Chai Wah Wu, Dec 20 2025

Crossrefs

Cf. A131574, A264499.

Sequence in context: A097102 A281877 A159781 * A102924 A214017 A083738

Adjacent sequences: A264495 A264496 A264497 * A264499 A264500 A264501

Keyword

nonn

Author

Colin Barker, Nov 15 2015

Status

approved