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

1,2

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

Links

Table of n, a(n) for n=1..9.

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

Formula

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

Example

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

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 *)

Prog

(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)))
);

Crossrefs

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

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

Sequence in context: A301539 A240517 A074022 * A157835 A071632 A259427

Adjacent sequences: A232445 A232446 A232447 * A232449 A232450 A232451

Keyword

nonn,more,hard

Author

Stanislav Sykora, Nov 24 2013

Extensions

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

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

Status

approved