A232449 -id:A232449 - cp's OEIS frontend

A232448 Indices of Belphegor primes: numbers k such that the decimal number 1 0...0(k 0's) 666 0...0(k 0's) 1 (i.e., A232449(k)) is prime.

0, 13, 42, 506, 608, 2472, 2623, 28291, 181298

Offset: 1

Stanislav Sykora, Nov 24 2013

The resulting primes might be called Belphegor primes, after Pickover (see link). - N. J. A. Sloane, Dec 14 2015
I suspect the larger numbers only correspond to probable primes. - N. J. A. Sloane, Oct 16 2018
The numbers correspond to proven primes for n <= 9. - Jens Kruse Andersen, Mar 25 2021

			0 is in the sequence because A232449(0) = 16661 is prime.
13 is in the sequence because A232449(13) = 1000000000000066600000000000001 is prime.
For k = 1..12, A232449(k) is composite.
42 is in the sequence because 10000000000000000000000000000000000000000006660000000000000000000000000000\
000000000000001 is a (probable) prime. - _N. J. A. Sloane_, Oct 16 2018

Tony Padilla and Brady Haran, The Most Evil Number, Numberphile video (2018)
Clifford A. Pickover, Belphegor's Prime: 1000000000000066600000000000001
Simon Singh, Homer Simpson's scary math problems. BBC News. Retrieved 31 October 2013.
Eric Weisstein's World of Mathematics, Belphegor Prime
Eric Weisstein's World of Mathematics, Integer Sequence Primes
Wikipedia, Belphegor's prime

Cf. A232449 (Belphegor numbers), A232450, A232451.

Cf. A156166 (= a(n) + 1).

lst = {}; Do[p = 10^(2*n + 4) + 666*10^(n + 1) + 1; If[PrimeQ[p], Print[n]], {n, 0, 3000}]; (* Nathaniel Johnston, Nov 25 2013 *)

default(factor_proven,1);
Belphegor(k)=(10^(k+3)+666)*10^(k+1)+1;
for (an=0,10000,
  if (isprime(Belphegor(an)),print("Found: ",an),
      if (an%100==0,print("Tested up to: ",an)))
);

a(n) = A156166(n) - 1.

a(9) based on A156166 from Eric W. Weisstein, Jan 24 2017

Offset changed to 1 by Jon E. Schoenfield, Mar 23 2021

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

Original entry on oeis.org

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

Stanislav Sykora, Nov 24 2013

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)

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

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

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

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

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

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

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

A232450 Largest prime factor of the Belphegor number B(n) = (10^(n+3) + 666)*10^(n+1) + 1.

Original entry on oeis.org

16661, 1103, 1417831, 1143749, 14282381, 11699423, 1950071, 7503425119, 3837692792387, 145857793, 76607717987, 1755833757671518620617, 17416012536871141, 1000000000000066600000000000001, 16540928199996367, 744657085412168192717253704669

Offset: 0

Stanislav Sykora, Nov 24 2013

nonn

The Belphegor numbers (A232449), though not often prime themselves (see A232448), tend to contain very large prime factors and are therefore hard to factorize.

Stanislav Sykora and Amiram Eldar, Table of n, a(n) for n = 0..64 (terms 0..44 from Stanislav Sykora)
Clifford A. Pickover, Belphegor's Prime: 1000000000000066600000000000001
Wikipedia, Belphegor's prime

Cf. A232448 (indices of Belphegor primes), A232449 (Belphegor numbers).

Table[FactorInteger[(10^(n + 3) + 666)*10^(n + 1) + 1][[-1, 1]], {n, 20}] (* T. D. Noe, Nov 25 2013 *)

default(factor_proven,1);
Belphegor(k)=(10^(k+3)+666)*10^(k+1)+1;
LargestPrimeFactor(k)={local(f);f=factor(k);return(f[#f[,1],1])};
nmax=40; v=vector(nmax);
for (n=0,#v-1,v[n+1]=LargestPrimeFactor(Belphegor(n));print(v[n+1]))

cp's OEIS Frontend

A232448 Indices of Belphegor primes: numbers k such that the decimal number 1 0...0(k 0's) 666 0...0(k 0's) 1 (i.e., A232449(k)) is prime.

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

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A232450 Largest prime factor of the Belphegor number B(n) = (10^(n+3) + 666)*10^(n+1) + 1.

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