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

Author

Stanislav Sykora, Nov 24 2013

Keywords

• nonn
• more
• hard

Comments

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

Examples

			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
		

Links

• 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

Crossrefs

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

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

Programs

Mathematica

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 *)

PARI

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)))
);

Formula

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

Extensions

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

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