A349708 a(n) is the smallest positive number k such that (product of the first n odd primes) + k^2 is a square.

1, 1, 4, 1, 19, 53, 58, 97, 181, 4244, 2122, 31126, 16451, 297392, 2444006, 622249, 2909047, 216182072, 62801719, 769709491, 32522441312, 37859955467, 129549407177, 286721160343, 101419856449, 107709289064864, 72441253480727, 56099073382147, 5249126879235893

Offset: 1

Author

Richard Peterson, Dec 31 2021

Keywords

• nonn

Comments

a(n) is half the difference between the middle two divisors of A070826(n + 1). - David A. Corneth, Jan 17 2022

Examples

			a(4)=1 because the product of the first 4 odd primes, 3*5*7*11 = 1155, is 34^2 - 1. a(5)=19 because 15015=3*5*7*11*13=124^2-19^2, and no positive integer less than 19 will work in this situation.
		

Links

• Jon E. Schoenfield, Table of n, a(n) for n = 1..42

Crossrefs

Cf. A056737, A061060, A070826, A236381.

Programs

PARI

a(n) = my(k=1, p=prod(k=2, n+1, prime(k))); while (!issquare(k^2+p), k++); k; \\ Michel Marcus, Jan 10 2022

Python

from math import isqrt
from sympy import primorial, divisors
def A349708(n):
    m = primorial(n+1)//2
    a = isqrt(m)
    d = max(filter(lambda d: d <= a, divisors(m,generator=True)))
    return (m//d-d)//2 # Chai Wah Wu, Mar 29 2022

Extensions

a(15)-a(26) and corrections to a(9) and a(11) from Jinyuan Wang, Jan 07 2022

a(27)-a(30) from Jon E. Schoenfield, Jan 16 2022