cp's OEIS Frontend

The corresponding values of k are 2, 3, 6, 15, 715 = A215659.

The equation p# + k = k^2 has an integer solution k if and only if 1 + 4*p# is a square.

Conjecture: Not the same sequence as A192579, which is finite.

When p is in this sequence, p# = k(k-1) is in A161620, the intersection of A002110 and A002378. - Jeppe Stig Nielsen, Mar 27 2018

Gica proved that if p is a prime different from 2, 3, 5, 7, 13, 37, then there exists a prime q < p which is a quadratic residue modulo p and q == 1 (mod 4).

Gica proved that if p is a prime different from 2, 3, 5, 7, 11, 13, then a prime q < p exists which is a quadratic residue modulo p and q == 1 (mod 3). His paper has not yet been published, but see A192578 for a reference, link, and examples of a similar result.

Gica proved that if p is a prime different from 2, 3, 5, 13, then a prime q < p exists which is a quadratic residue modulo p and q == 2 (mod 3). His paper has not yet been published, but see A192578 for a reference, link, and examples of a similar result.

In Chapter 7 of de Weger's tract, it is shown that there are no other terms.

More generally, de Weger exposited how one can determine all squares which can be represented as the sum or difference of k-smooth numbers for any given k and determined all integers whose squares can be represented as the sum or difference of 7-smooth numbers, among which the largest one is 14117^2 = 199289869 = 3^13 * 5^3 - 2 * 7^3.