A239669 Total number of prime factors counted with multiplicity of prime(n)-1 and prime(n)+1, where prime(n) is the n-th prime.

1, 3, 4, 5, 5, 5, 7, 6, 6, 6, 8, 6, 7, 6, 7, 7, 6, 6, 6, 8, 7, 8, 6, 8, 9, 7, 7, 7, 8, 8, 11, 7, 7, 7, 7, 8, 6, 8, 7, 6, 7, 8, 10, 9, 8, 9, 7, 9, 6, 7, 8, 9, 9, 9, 11, 7, 8, 10, 6, 8, 6, 7, 8, 8, 7, 6, 7, 9, 6, 8, 9, 8, 8, 7, 9, 10, 7, 7, 9, 8, 8, 7, 10, 10

Offset: 1

Author

Chris Boyd, Mar 23 2014

Keywords

• nonn

Comments

If a(n) <= 6, n is a distinguished prime (A106639). According to a conjecture in A106639, 6 is the least term appearing infinitely often. Numerical evidence suggests the conjecture that other - perhaps all - integers > 6 appear infinitely often. The strong conjecture implies that all terms but one in A155800 are positive.

Examples

			a(7) = 7 as prime(7) = 17, 17-1 = 2^4 and 17+1 = 2*3^2, so total number of prime factors = 4+3 = 7.
		

Links

• Chris Boyd, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000040, A106639, A155800.

Programs

PARI

forprime(p=1,500,print1(bigomega(p-1) + bigomega(p+1)", "))

Formula

a(n) = Omega(prime(n)^2 - 1), where Omega is A001222.