A232451 Number of prime divisors of (10^(n+3) + 666)*10^(n+1) + 1 (see A232449) counted with multiplicity.

1, 2, 2, 3, 4, 3, 5, 4, 2, 5, 5, 4, 3, 1, 2, 3, 6, 4, 3, 6, 4, 2, 4, 5, 2, 4, 3, 6, 7, 7, 4, 3, 2, 4, 5, 3, 4, 7, 4, 6, 6, 4, 1, 4, 5, 4, 6, 6, 5, 3, 6, 4, 6, 6, 4, 11, 6, 6, 6, 4, 5, 5, 2, 6, 7

Offset: 0

Author

Stanislav Sykora, Nov 24 2013

Keywords

• nonn
• more

Comments

The Belphegor numbers (A232449), though large and rarely prime (A232448), tend to contain only very few prime factors. One wonders whether this sequence might be bounded.

From Robert Israel, Feb 23 2017: (Start)

The sequence is unbounded.

Indeed, if p is in A001913 such that the polynomial 10^4 x^2 + 6660 x + 1 has a simple root mod p, then for all k there exist Belphegor numbers divisible by p^k.

For example, p=29 works; we have A232449(n) divisible by 29^k for n = 6, 158, 5522, 41570, 8153130, 107172470, 3553045502, 136793469406, 2761185750502, 142830181379582, ...

(End)

Links

• FactorDB, (10^(n+3)+666)*10^(n+1)+1.
• Clifford A. Pickover, Belphegor's Prime: 1000000000000066600000000000001
• Wikipedia, Belphegor's prime

Crossrefs

Cf. A001913, A232448 (indices of Belphegor primes), A232449 (Belphegor numbers), A232450 (largest prime factor of A232449(n)).

Programs

Magma

[&+[p[2]: p in Factorization(666*10^(n+1)+100^(n+2)+1)]: n in [0..40]]; // Bruno Berselli, Nov 27 2013

Maple

seq(numtheory:-bigomega(10^(2*n+4)+666*10^(n+1)+1), n=0..30); # Robert Israel, Feb 23 2017

Mathematica

Table[Total[Transpose[FactorInteger[(10^(n + 3) + 666)*10^(n + 1) + 1]][[2]]], {n, 0, 25}] (* T. D. Noe, Nov 28 2013 *)

PARI

a(n)=bigomega(10^(n+1)*(10^(n+3)+666)+1) \\ Charles R Greathouse IV, Nov 26 2013

Extensions

a(45)-a(64) from Amiram Eldar, Apr 11 2020