A349708 - OEIS

%I #70 Mar 31 2022 02:49:50

%S 1,1,4,1,19,53,58,97,181,4244,2122,31126,16451,297392,2444006,622249,

%T 2909047,216182072,62801719,769709491,32522441312,37859955467,

%U 129549407177,286721160343,101419856449,107709289064864,72441253480727,56099073382147,5249126879235893

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

%C a(n) is half the difference between the middle two divisors of A070826(n + 1). - _David A. Corneth_, Jan 17 2022

%H Jon E. Schoenfield, <a href="/A349708/b349708.txt">Table of n, a(n) for n = 1..42</a>

%e 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.

%o (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

%o (Python)

%o from math import isqrt

%o from sympy import primorial, divisors

%o def A349708(n):

%o     m = primorial(n+1)//2

%o     a = isqrt(m)

%o     d = max(filter(lambda d: d <= a, divisors(m,generator=True)))

%o     return (m//d-d)//2 # _Chai Wah Wu_, Mar 29 2022

%Y Cf. A056737, A061060, A070826, A236381.

%K nonn

%O 1,3

%A _Richard Peterson_, Dec 31 2021

%E a(15)-a(26) and corrections to a(9) and a(11) from _Jinyuan Wang_, Jan 07 2022

%E a(27)-a(30) from _Jon E. Schoenfield_, Jan 16 2022