Euler’s lucky number

A prime number $p$ is one of Euler’s lucky numbers if $n^2-n+p$ for each is also a prime. Put another way, a lucky number of Euler’s plus the $n$th oblong number produces a list of primes $p$-long. There are only six of them: 2, 3, 5, 11, 17 and 41, these are listed in A014556 of Sloane’s OEIS.

41 is perhaps the most famous of these. We can verify that 2 + 41 is 43, a prime, that 47 is also prime, so are 53, 61, 71, 83, 97, and so on to 1601, giving a list of 41 primes. Predictably, 1681 is divisible by 41, being its square. For $n>p$ the formula does not consistently give only composites or only primes.

Title	Euler’s lucky number
Canonical name	EulersLuckyNumber
Date of creation	2013-03-22 16:55:33
Last modified on	2013-03-22 16:55:33
Owner	PrimeFan (13766)
Last modified by	PrimeFan (13766)
Numerical id	4
Author	PrimeFan (13766)
Entry type	Definition
Classification	msc 11A41
Synonym	lucky number of Euler
Synonym	Eulerian lucky number