A232448 -id:A232448 - cp's OEIS frontend

A131645 Beastly primes (version 2): primes containing 666 as a substring.

6661, 16661, 26669, 46663, 56663, 66601, 66617, 66629, 66643, 66653, 66683, 66697, 76667, 96661, 96667, 106661, 106663, 106669, 116663, 146669, 166601, 166603, 166609, 166613, 166619, 166627, 166631, 166643, 166657, 166667, 166669, 166679

Offset: 1

Tanya Khovanova, Sep 08 2007

These are the primes among the beastly numbers A051003.
There are several other definitions of beastly primes (see cross-references).
Asymptotic density n/log(n), since almost all primes are of this form.

Charles R Greathouse IV and Harry J. Smith, Table of n, a(n) for n = 1..20000
Tony Padilla and Brady Haran, The Most Evil Number, Numberphile video (2018)

Cf. A051003, A186086, A046720, A138563, A232448, A321001.

Select[Range[300000], StringFreeQ[ToString[ # ], "666"] == False && PrimeQ[ # ] &]
Select[Prime[Range[300000]],!StringFreeQ[ToString[ # ],"666"]&] (* Zak Seidov, Jan 09 2009 *)

digitsIn(x) = 1 + log(x)\log(10)
allocatemem(932245000);
default(primelimit, 4294965247); m=1; forprime (p=6660, 68466670, d=digitsIn(p); for (i=1, d-3, t=10^i; u=p\t; x=u-(u\1000)*1000; if (x==666, print(m, " ", p); write("b131645.txt", m, " ", p); m++; break))) \\ Harry J. Smith, Jan 11 2009

a(n) ~ n log n. - Charles R Greathouse IV, Oct 13 2015

Definition corrected by Arkadiusz Wesolowski, Feb 12 2011

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

A156166 Numbers k > 0 such that (10^(k+2) + 666)*10^k + 1 is prime.

Original entry on oeis.org

1, 14, 43, 507, 609, 2473, 2624, 28292, 181299

Offset: 1

M. F. Hasler, Feb 10 2009

Or, indices of primes in the sequence 16661, 1066601, 100666001, 10006660001,... Ondrejka calls these "beastly palindromic primes".

In popular culture: the number generated by a(2), 1000000000000066600000000000001, also known as Belphegor's Prime, was used as a plot device in Episode "Just a Regular Irregular" of the "Elementary" TV series (first aired Nov/13/2014). - Serge Batalov, Nov 15 2014

C. Caldwell, H. Dubner (Eds): The top ten prime numbers: from the unpublished collections of R. Ondrejka (May 2001), p. 32
Internet Movie Database, Elementary: Season 3, Episode 3: Just a Regular Irregular
Clifford A. Pickover, Belphegor's Prime: 1000000000000066600000000000001
Eric Weisstein's World of Mathematics, Belphegor Prime
Eric Weisstein's World of Mathematics, Integer Sequence Primes
Wikipedia, Belphegor's prime

Cf. A082700 and search results for 16661.

Cf. A232448 (a(n) - 1).

[n: n in [1..500] | IsPrime((10^(n+2)+666)*10^n+1)]; // Vincenzo Librandi, Nov 15 2014

A156166:=n->`if`(isprime((10^(n+2)+666)*10^n+1), n, NULL): seq(A156166(n), n=1..10^3); # Wesley Ivan Hurt, Nov 16 2014

Select[Range[10^3], PrimeQ[(10^(# + 2) + 666)*10^# + 1] &] (* Arkadiusz Wesolowski, Sep 08 2011 *)

for( n=1,9999, ispseudoprime((10^(n+2)+666)*10^n+1) & print1(n","))

a(n) = A232448(n) + 1.

a(8) = 28292 (discovered on Jan 05 2004, by Daniel Heuer), Arkadiusz Wesolowski, Mar 16 2011

a(9) = 181299 from Serge Batalov, Nov 15 2014

A232449 The palindromic Belphegor numbers: (10^(n+3)+666)*10^(n+1)+1.

Original entry on oeis.org

16661, 1066601, 100666001, 10006660001, 1000066600001, 100000666000001, 10000006660000001, 1000000066600000001, 100000000666000000001, 10000000006660000000001, 1000000000066600000000001, 100000000000666000000000001, 10000000000006660000000000001, 1000000000000066600000000000001

Offset: 0

Stanislav Sykora, Nov 24 2013

Though this sequence rarely contains primes (see A232448), most of its members tend to contain a few very large prime factors. The name stems from 'Belphegor's Prime', a(13), which was so named by Clifford Pickover (see link). [Comment corrected by N. J. A. Sloane, Dec 14 2015]

Stanislav Sykora, Table of n, a(n) for n = 0..497
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 Number
Wikipedia, Belphegor's prime
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A232448, A232450, A232451.

Subsequence of A118598.

A232449[n_] := 100^(n+2) + 666*10^(n+1) + 1; Array[A232449, 15, 0] (* or *)
LinearRecurrence[{111, -1110, 1000}, {16661, 1066601, 100666001}, 15] (* Paolo Xausa, Mar 27 2025 *)

Belphegor(k)=(10^(k+3)+666)*10^(k+1)+1; nmax = 498; v = vector(nmax); for (n=0,#v-1, v[n+1]=Belphegor(n))

a(n) = 666*10^(n+1)+100^(n+2)+1.

G.f.: (16661 - 782770*x + 767000*x^2) / ((1 - x)*(1 - 10*x)*(1 - 100*x)). [Bruno Berselli, Nov 25 2013]

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]))

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

A321001 Primes which contain the fax number of the beast (667).

Original entry on oeis.org

1667, 6673, 6679, 10667, 15667, 16673, 25667, 31667, 34667, 36671, 36677, 39667, 42667, 45667, 46679, 49667, 52667, 54667, 55667, 56671, 57667, 61667, 63667, 64667, 66701, 66713, 66721, 66733, 66739, 66749, 66751, 66763, 66791, 66797, 70667, 76667

Offset: 1

N. J. A. Sloane, Nov 01 2018

Since the so-called beastly primes (containing the digit string 666) were mentioned recently in a Numberphile video (see link), I guess it is now OK to include these numbers (which are based on the old convention that your fax number was one more than your telephone number) in the OEIS.

Note that there is no analog of A232448, because 1 0^k 667 0^k 1 is always divisible by 3 and is never prime.

Tony Padilla and Brady Haran, The Most Evil Number, Numberphile Video, October 2018
Margaret Wertheim, The Fax Numbers of the Beast, and Other Mathematical Sports: An Interview with Neil Sloane, Cabinet Magazine, Issue 57, Spring 2015, pages 48-54.

Cf. beastly primes A131645, also A232448.

cp's OEIS Frontend

A131645 Beastly primes (version 2): primes containing 666 as a substring.

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A156166 Numbers k > 0 such that (10^(k+2) + 666)*10^k + 1 is prime.

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A232449 The palindromic Belphegor numbers: (10^(n+3)+666)*10^(n+1)+1.

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

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Magma

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A321001 Primes which contain the fax number of the beast (667).

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