A239669 - OEIS

%I #13 Aug 05 2014 02:40:28

%S 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,

%T 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,

%U 9,6,8,9,8,8,7,9,10,7,7,9,8,8,7,10,10

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

%C 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.

%H Chris Boyd, <a href="/A239669/b239669.txt">Table of n, a(n) for n = 1..10000</a>

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

%e 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.

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

%Y Cf. A000040, A106639, A155800.

%K nonn

%O 1,2

%A _Chris Boyd_, Mar 23 2014