The Pentagonal Numbers and their Link to an Integer Sequence which contains the Primes of Form 6n-1

A binary operation is a calculus that combines two elements to obtain another elements. It seems quite simple for numbers, because we usually imagine it as a simple sum or product. However, also in the case of numbers, a binary operation can be extremely fascinating if we consider it in a generalized form. Previously, several examples have been proposed of generalized sums for different sets of numbers (Fibonacci, Mersenne, Fermat, q-numbers, repunits and others). These sets can form groupoid which possess different binary operators. Here we consider the pentagonal numbers (OEIS A000326). We will see the binary operation for them. This generalized sum involves another integer sequence, OEIS A016969, and this sequence contains OEIS A007528, that is the primes of the 6n-1.