A057934 Number of prime factors of 10^n + 1 (counted with multiplicity).

1, 1, 3, 2, 2, 2, 2, 2, 5, 3, 5, 3, 3, 4, 7, 5, 4, 3, 2, 4, 8, 4, 5, 3, 5, 3, 7, 4, 3, 7, 2, 4, 9, 4, 5, 6, 4, 3, 10, 4, 3, 7, 4, 4, 12, 4, 4, 9, 4, 7, 8, 4, 2, 6, 10, 5, 6, 5, 4, 6, 3, 3, 12, 3, 6, 8, 2, 4, 10, 11, 3, 5, 4, 7, 11, 6, 12, 7, 4, 9, 11, 3, 7, 8, 8, 3, 8, 4, 4, 11, 6, 4, 8, 4, 6, 8, 4, 5, 13

Offset: 1

Author

Patrick De Geest, Oct 15 2000

Keywords

• nonn

Comments

2^(a(2n)-1)-1 predicts the number of pair-solutions of even length L for AB = A^2 + B^2. For instance, with length 18 we have 10^18 + 1 = 101*9901*999999000001 or 3 divisors F which when put into the Mersenne formula 2^(F-1)-1 yields 3 pairs (see reference 'Puzzle 104' for details).

Links

• Max Alekseyev, Table of n, a(n) for n = 1..331 (first 310 terms from Ray Chandler)
• Makoto Kamada, Factorizations of 100...001.
• Carlos Rivera, Puzzle 104, The Prime Puzzles & Problems Connection.
• S. S. Wagstaff, Jr., Main Tables from the Cunningham Project.
• S. S. Wagstaff, Jr., The Cunningham Project

Crossrefs

bigomega(b^n+1): this sequence (b=10), A057935 (b=9), A057936 (b=8), A057937 (b=7), A057938 (b=6), A057939 (b=5), A057940 (b=4), A057941 (b=3), A054992 (b=2).

Cf. A003021, A001271, A046053, A001562, A057951, A119704, A344897.

Programs

Mathematica

PrimeOmega[10^Range[100]+1] (* Harvey P. Dale, May 02 2021 *)

PARI

a(n)=bigomega(10^n+1) \\ Charles R Greathouse IV, Sep 14 2015

Formula

a(n) = A057951(2n) - A057951(n). - T. D. Noe, Jun 19 2003