A110874 Number of prime factors of 2 + n^(n+1) counted with multiplicity.

1, 2, 1, 5, 2, 2, 4, 5, 2, 5, 4, 4, 5, 3, 1, 4, 5, 3, 4, 6, 3, 8, 4, 5, 4, 4, 2, 6, 3, 6, 5, 5, 5, 6, 6, 8, 6, 6, 4, 5, 4, 6, 4, 5, 3, 8, 4, 3, 5, 5, 5, 7, 7, 11, 4, 5, 4, 13, 4, 6, 2, 5, 2, 6, 6, 5, 8, 9, 5, 9, 4, 7, 4, 4, 5, 7, 6, 7, 6, 9, 4, 9, 5, 8, 5, 8

Offset

1,2

Comments

Compared with A110676, number of prime factors with multiplicity of 2 + n^(n+1), this seems to have an unlimited number of primes (n = 1, 3, 15, ...) and semiprimes (n = 2, 5, 6, 9, 27, ...). Of course, n even gives n | a(n).

Links

Sean A. Irvine, Table of n, a(n) for n = 1..100

Formula

a(n) = A001222(1 + A110567(n)) = A001222(2 + A007778(n)) = A001222(2 + n^(n+1)).

Example

a(1) = 1 because 2 + 1^2 = 3 is prime (one prime factor).
a(2) = 2 because 2 + 2^3 = 10 = 2 * 5 is semiprime (two prime factors).
a(3) = 1 because 2 + 3^4 = 83 is prime.
a(4) = 5 because 2 + 4^5 = 1026 = 2 * 3^3 * 19 has five prime factors (3 has multiplicity of 3).
a(5) = 2 because 2 + 5^6 = 15627 = 3 * 5209 is semiprime (two prime factors).
a(6) = 2 because 2 + 6^7 = 279938 = 2 * 139969 is semiprime (two prime factors).
a(15) = 1 because 2 + 15^16 = 6568408355712890627 is prime. What is the next prime?

Mathematica

Table[PrimeOmega[2+n^(n+1)], {n, 41}] (* Harvey P. Dale, Nov 08 2020 *)

Crossrefs

Cf. A001222, A007778, A110567, A110676.

Sequence in context: A308569 A350073 A174978 * A010253 A065274 A260325

Adjacent sequences: A110871 A110872 A110873 * A110875 A110876 A110877

Keyword

nonn

Author

Jonathan Vos Post, Sep 18 2005

Extensions

More terms from Sean A. Irvine, Sep 17 2023

Status

approved