A194003 Number of prime factors of n^8 + 1, counted with multiplicity.

0, 1, 1, 3, 1, 3, 2, 3, 3, 2, 2, 3, 3, 2, 3, 3, 2, 3, 2, 3, 2, 3, 2, 4, 3, 3, 2, 6, 2, 4, 3, 3, 2, 2, 2, 4, 3, 3, 2, 4, 6, 3, 2, 2, 4, 3, 3, 2, 3, 3, 2, 2, 2, 2, 3, 3, 2, 5, 2, 3, 2, 4, 4, 4, 3, 6, 2, 5, 2, 2, 2, 5, 2, 5, 4, 4, 3, 4, 3, 5, 4, 2, 3, 4, 2, 4

Offset

0,4

Comments

This is to A193330 as A002523(n) = n^4+1 is to A002522(n) = n^2 + 1, and as A060890(n) = n^8+1 is to A002522(n) = n^2 + 1. a(n) = 1 when n^8+1 is prime, iff n is in {1, 2, 4} unless there is a larger Fermat prime than 65537.

Links

Amiram Eldar, Table of n, a(n) for n = 0..10000

Formula

a(n) = A001222(A060890(n)) = bigomega(n^8+1) or Omega(n^8+1)

Example

a(10) = 2 because 10^8 + 1 = 100000001 = 17 * 5882353 has 2 prime factors.
a(40) = 6 because 40^8 + 1 = 6553600000001 = 17^2 * 113 * 337 * 641 * 929 has 6 prime factors (with multiplicity) and is the smallest example not squarefree.

Mathematica

Join[{0}, Table[Total[Transpose[FactorInteger[n^8 + 1]][[2]]], {n, 50}]]
PrimeOmega[Range[0, 90]^8+1] (* Harvey P. Dale, May 27 2018 *)

Prog

(PARI) a(n) = bigomega(n^8+1); \\ Michel Marcus, Feb 09 2020
(Magma) [0] cat [&+[p[2]: p in Factorization(n^8+1)]:n in [1..90]]; // Marius A. Burtea, Feb 09 2020

Crossrefs

Cf. A001222, A060890, A002523, A193432, A193562, A193929.

Sequence in context: A178055 A247856 A059660 * A339365 A035456 A035664

Adjacent sequences: A194000 A194001 A194002 * A194004 A194005 A194006

Keyword

nonn,easy

Author

Jonathan Vos Post, Aug 10 2011

Status

approved