A156166 -id:A156166 - cp's OEIS frontend

A186086 Beastly primes (version 1): either 666 followed by 0's and a 1 or 7 at the right end or a palindrome with 666 in the center, 0's surrounding these digits, and 1 or 7 at both ends.

6661, 16661, 66601, 76667, 700666007, 6660000000001, 666000000000001, 700000666000007, 70000006660000007, 6660000000000000000000000007, 66600000000000000000000000007, 1000000000000066600000000000001

Offset: 1

Arkadiusz Wesolowski, Feb 12 2011

Differs from A131645 in that 26669, 46663, 56663, 66617, 66629, 66643, 66653, 66683, 66697, 96661, 96667, 106661, 106663, 106669, 116663, 146669, 166601, 166603, 166609, 166613, 166619, 166627, 166631, 166643, 166657, 166667, 166669, 166679, are not included here.

76667 is the largest beastly prime that does not contain the digit "0".

Charles R Greathouse IV, Table of n, a(n) for n = 1..33
Chris Caldwell, The Prime Glossary, Beastly prime
G. L. Honaker, Jr. and Chris Caldwell, Prime Curios! 6661
R. Ondrejka, The Top Ten: a Catalogue of Primal Configurations (May 2001), p. 5
Tony Padilla and Brady Haran, The Most Evil Number, Numberphile video (2018)
Eric Weisstein's World of Mathematics, Beast Number

Cf. A046720, A051003, A131645, A164968, A196023, A138563, A186630, A186538, A156166, A186521.

e = 14; p = 666*10^n + 1; q = (10^(n + 2) + 666)*10^n + 1; Select[Union[Table[p, {n, 2*e}], Table[p + 6, {n, 2*e}], Table[q, {n, e}], Table[q + 6*10^(2*n + 2) + 6, {n, e}]], PrimeQ] (* Arkadiusz Wesolowski, Sep 21 2011 *)
Module[{nn=35,bp1,bp2,bp3,bp4}, bp1=FromDigits/@ Table[Join[PadRight[ {6,6,6},n1,0],{1}],{n1,3,nn}]; bp2=FromDigits/@ Table[Join[ PadRight[ {6,6,6},n2,0],{7}], {n2,3,nn}]; bp3=FromDigits/@ Table[Join[PadRight[ {1},n3,0], {6,6,6},PadLeft[ {1},n3,0]],{n3,1,nn/2}];bp4=FromDigits/@ Table[Join[PadRight[{7},n3,0],{6,6,6}, PadLeft[ {7},n3,0]],{n3,1,nn/2}]; Select[Sort[Join[bp1,bp2,bp3,bp4]],PrimeQ]] (* Harvey P. Dale, Jan 18 2017 *)

Edited by N. J. A. Sloane, Feb 12 2011

a(10)-a(12) from Charles R Greathouse IV, Feb 12 2011

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.

Original entry on oeis.org

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

A246804 Numbers k such that (10^(k+2) + 999) * 10^k + 1 is prime.

Original entry on oeis.org

1, 3, 15, 135, 645, 1373, 195317, 237249

Offset: 1

Serge Batalov, Nov 16 2014

Or, indices of primes in the sequence of decimal palindromes 19991, 1099901, 100999001, 10009990001, ...

Or, numbers k such that there exists an "upside-down-Belphegor's primes" of length 2*k+3.

Cf. A156166 (Belphegor's primes), A082703 (plateau primes 199...991).

[n: n in [1..500] | IsPrime((10^(n+2)+999)*10^n+1)];

A246804:=n->`if`(isprime((10^(n+2)+999)*10^n+1), n, NULL): seq(A246804(n), n=1..10^3); # Wesley Ivan Hurt, Nov 16 2014

Select[Range[10^3], PrimeQ[(10^(# + 2) + 999)*10^# + 1] &]

for( n=1,9999, ispseudoprime((10^(n+2)+999)*10^n+1) & print1(n","))

cp's OEIS Frontend

A186086 Beastly primes (version 1): either 666 followed by 0's and a 1 or 7 at the right end or a palindrome with 666 in the center, 0's surrounding these digits, and 1 or 7 at both ends.

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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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A246804 Numbers k such that (10^(k+2) + 999) * 10^k + 1 is prime.

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