A193929 -id:A193929 - cp's OEIS frontend

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

Jonathan Vos Post, Aug 10 2011

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.

			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.

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

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

[0] cat [&+[p[2]: p in Factorization(n^8+1)]:n in [1..90]]; // Marius A. Burtea, Feb 09 2020

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

a(n) = bigomega(n^8+1); \\ Michel Marcus, Feb 09 2020

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

A193759 Array, by antidiagonals, A(k,n) is the number of prime factors of n^(2^k) + 1, counted with multiplicity.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, 1, 1, 2, 0, 1, 1, 2, 1, 0, 1, 1, 3, 1, 2, 0, 1, 2, 2, 1, 2, 1, 0, 1, 2, 2, 2, 3, 1, 3, 0, 1, 2, 2, 2, 3, 2, 2, 2, 0, 1, 2, 6, 2, 4, 3, 3, 2, 2, 0, 1, 3, 5, 2, 4, 3, 3, 3, 3, 1, 3, 0, 1, 4, 7, 3, 4, 3, 4, 3, 2, 2, 2, 1, 0, 1, 5

Offset: 0

Jonathan Vos Post, Aug 11 2011

The main diagonal A(n,n) = number of prime factors of n^(2^n) + 1, counted with multiplicity, begins 0, 1, 1, 3, 2, 4, 3, 6, 6.

			A(4,5) = 3 because 1+5^16 = 152587890626 = 2 * 2593 * 29423041, which has 3 prime factors. The array begins:
================================================================
....|n=0|n=1|n=2|n=3|n=4|n=5|n=6|n=7|n=8|n=9|.10|.11|comment
====|===|===|===|===|===|===|===|===|===|===|===|===|===========
k=0.|.0.|.1.|.1.|.2.|.1.|.2.|.1.|.3.|.2.|.2.|.1.|.3.|A001222
k=1.|.0.|.1.|.1.|.2.|.1.|.2.|.1.|.3.|.2.|.2.|.1.|.2.|A193330
k=2.|.0.|.1.|.1.|.2.|.1.|.2.|.1.|.2.|.2.|.3.|.2.|.2.|A193929
k=3.|.0.|.1.|.1.|.3.|.1.|.3.|.2.|.3.|.3.|.2.|.2.|.3.|A194003
k=4.|.0.|.1.|.1.|.2.|.2.|.3.|.3.|.3.|.3.|.2.|.5.|.3.|not in OEIS
k=5.|.0.|.1.|.2.|.2.|.2.|.4.|.3.|.4.|.3.|.2.|.4.|.4.|not in OEIS
================================================================

Cf. A001222, A002523, A060890, A193330, A193929, A194003.

Edited by Alois P. Heinz, Aug 11 2011

More terms from Max Alekseyev, Sep 09 2011

cp's OEIS Frontend

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

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