A051003 -id:A051003 - cp's OEIS frontend

A002282 a(n) = 8*(10^n - 1)/9.

0, 8, 88, 888, 8888, 88888, 888888, 8888888, 88888888, 888888888, 8888888888, 88888888888, 888888888888, 8888888888888, 88888888888888, 888888888888888, 8888888888888888, 88888888888888888, 888888888888888888, 8888888888888888888, 88888888888888888888, 888888888888888888888

Offset: 0

N. J. A. Sloane

If the initial term is omitted, might be called eightful (or hateful) numbers!

			Curious multiplications:
9*9 + 7 = 88;
98*9 + 6 = 888;
987*9 + 5 = 8888;
9876*9 + 4 = 88888;
98765*9 + 3 = 888888;
987654*9 + 2 = 8888888;
9876543*9 + 1 = 88888888;
98765432*9 + 0 = 888888888;
987654321*9 - 1 = 8888888888;
9876543210*9 - 2 = 88888888888. - _Philippe Deléham_, Mar 09 2014

Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, page 32.

Harry J. Smith, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (11,-10).

Cf. A002275, A002276, A002277, A002278, A002279, A002280, A002281, A002283, A059988, A075412, A178635.

Cf. A010785, A017257, A051003, A059482, A246058.

A002282:=n->8*(10^n - 1)/9; seq(A002282(n), n=0..20); # Wesley Ivan Hurt, Mar 10 2014

LinearRecurrence[{11,-10}, {0,8}, 20] (* Harvey P. Dale, May 30 2013 *)

{ a=-4/5; for (n = 0, 200, a+=8*10^(n - 1); write("b002282.txt", n, " ", a); ) } \\ Harry J. Smith, Jun 27 2009

def a(n): return 8*(10**n - 1)//9 # Martin Gergov, Oct 19 2022

From Jaume Oliver Lafont, Feb 03 2009: (Start)
a(n) = 11*a(n-1) - 10*a(n-2), with a(0)=0, a(1)=8.
G.f.: 8*x/((1-x)*(1-10*x)). (End)
a(n) = A178635(n)/A002283(n). - Reinhard Zumkeller, May 31 2010
a(n) = a(n-1) + 8*10^(n-1), with a(0)=0. - Vincenzo Librandi, Jul 22 2010
a(n) = 8*A002275(n) = A002283(n) - A002275(n). - Carauleanu Marc, Sep 03 2016
From Ilya Gutkovskiy, Sep 03 2016: (Start)
E.g.f.: 8*(exp(9*x) - 1)*exp(x)/9.
a(n) = floor(8*10^n/9). (End)
From Elmo R. Oliveira, Jul 20 2025: (Start)
a(n) = (A246058(n) - 1)/2.
a(n) = A010785(A017257(n-1)) for n >= 1. (End)

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

Original entry on oeis.org

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

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.

Original entry on oeis.org

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

A046720 Subsequence of beastly primes (A186086) that are palindromes that begin and end with 7.

Original entry on oeis.org

76667, 700666007, 700000666000007, 70000006660000007

Offset: 1

G. L. Honaker, Jr.

Next term is 7_{0}^48_666_{0}^48_7, containing 101 digits, and is too large to include here.

The number of digits in the terms is 2*A186521(n)+3: 5, 9, 15, 17, 101, 1159, 1589, 2647, 2787, 4787, 6135, 26961 (some correspond to probable primes). - Jens Kruse Andersen, Jul 13 2014

Jens Kruse Andersen, Table of n, a(n) for n = 1..7
R. Ondrejka, The Top Ten: a Catalogue of Primal Configurations
Tony Padilla and Brady Haran, The Most Evil Number, Numberphile video (2018)

Cf. A051003, A131645, A186086, A186521.

Select[Table[(7*10^(n + 2) + 666)*10^n + 7, {n, 7}], PrimeQ] (* Arkadiusz Wesolowski, Sep 08 2011 *)
Select[Table[With[{s=PadRight[{7},n,0]},FromDigits[Join[s,{6,6,6},Reverse[s]]]],{n,8}],PrimeQ] (* Harvey P. Dale, Aug 05 2024 *)

a(n) = (7*10^(k+2)+666)*10^k+7, where k = A186521(n). - Jens Kruse Andersen, Jul 13 2014

Definition revised by N. J. A. Sloane, Feb 14 2011

A138563 Beastly fax numbers: numbers containing the fax number of the Beast (667, one more than its regular number) in their decimal expansion.

Original entry on oeis.org

667, 1667, 2667, 3667, 4667, 5667, 6667, 6670, 6671, 6672, 6673, 6674, 6675, 6676, 6677, 6678, 6679, 7667, 8667, 9667, 10667, 11667, 12667, 13667, 14667, 15667, 16667, 16670, 16671, 16672, 16673, 16674, 16675, 16676, 16677, 16678

Offset: 1

N. J. A. Sloane, May 13 2007

The sum of the reciprocals of numbers not in this sequence is convergent. - Adam P. Goucher, Apr 27 2014

Jens Kruse Andersen, Table of n, a(n) for n = 1..1000
Robert Baillie and Thomas Schmelzer, Summing Kempner's Curious (Slowly-Convergent) Series, Mathematica Notebook kempnerSums.nb, Wolfram Library Archive, 2008.
Margaret Wertheim and Neil Sloane, The Fax Numbers of the Beast, and Other Mathematical Sports: An Interview with Neil Sloane, Cabinet, Spring 2015.

Cf. A051003.

Select[Range[20000], StringContainsQ[ToString[#], "667"] &] (* Amiram Eldar, Jun 28 2024 *)

a(n) ~ n. - Charles R Greathouse IV, Oct 25 2014

Sum_{k>=1, k is not a term} 1/k = 2301.846622336249707557560554200194249235044868457872023381489896767824372028... (calculated using Baillie and Schmelzer's kempnerSums.nb, see Links). - Amiram Eldar, Jun 28 2024

A098783 Smallest integer not occurring earlier such that the concatenation of the integers has the property that the first two digits sum to 9, the next two digits also, etc.

Original entry on oeis.org

0, 9, 1, 8, 2, 7, 3, 6, 4, 5, 18, 27, 36, 45, 54, 63, 72, 81, 90, 180, 91, 80, 92, 70, 93, 60, 94, 50, 95, 40, 96, 30, 97, 20, 98, 10, 909, 181, 82, 71, 83, 61, 84, 51, 85, 41, 86, 31, 87, 21, 88, 11, 809, 182, 73, 62, 74, 52, 75, 42, 76, 32, 77, 22, 78, 12, 709, 183, 64, 53, 65

Offset: 0

Eric Angelini, Oct 04 2004

This sequence contains no beastly numbers (A051003). - Rémy Sigrist, Dec 31 2017

			When parenthesis are added one can see the successive pairs of digits with sum 9: (0, 9), (1, 8), (2, 7), (3, 6), (4, 5), (18), (27), (36), (45), (54), (63), (72), (81), (90), (18)(0, 9)(1, ... - _Eric Angelini_, Dec 18 2017

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Rémy Sigrist, PARI program for A098783

Cf. A051003.

A244661 Beastly reciprocals, or numbers n such that digitsum(1/n) = 666.

Original entry on oeis.org

149, 298, 596, 646, 745, 1192, 1490, 1615, 2119, 2584, 2980, 3109, 3725, 3878, 5960, 6218, 6357, 6460, 7106, 7294, 7450, 8476, 9262, 9868, 10941, 11627, 11634, 11920, 12436, 14535, 14900, 15049, 15545, 16150, 18625, 21190, 22718, 23256, 23902, 24872, 24915

Offset: 1

Anthony Sand, Jul 04 2014

149 is a full reptend prime (see A001913), hence the sum of the decimal digits of 1/149 is 9 * 148 / 2 = 666.
From Robert G. Wilson v, Aug 16 2014: (Start)
If n is present, so is 10n.
If n is present then A003592*n is possibly present.
Primitives are: 149, 646, 1615, 2119, 3109, 3878, 7294, 9262, 9868, 10941, …, .
Palindromes: 646, 1525251, 2062602, …, .
Primes: 149, 3109, 111149, 351391, …, .
(End)

			If digitsum(1/n) sums the decimal digits of 1/n up to the point at which they recur or terminate, then digitsum(1/149) = 666 = 0 + 0 + 6 + 7 + 1 + 1 + 4 + 0 + 9 + 3 + 9 + 5 + 9 + 7 + 3 + 1 + 5 + 4 + 3 + 6 + 2 + 4 + 1 + 6 + 1 + 0 + 7 + 3 + 8 + 2 + 5 + 5 + 0 + 3 + 3 + 5 + 5 + 7 + 0 + 4 + 6 + 9 + 7 + 9 + 8 + 6 + 5 + 7 + 7 + 1 + 8 + 1 + 2 + 0 + 8 + 0 + 5 + 3 + 6 + 9 + 1 + 2 + 7 + 5 + 1 + 6 + 7 + 7 + 8 + 5 + 2 + 3 + 4 + 8 + 9 + 9 + 3 + 2 + 8 + 8 + 5 + 9 + 0 + 6 + 0 + 4 + 0 + 2 + 6 + 8 + 4 + 5 + 6 + 3 + 7 + 5 + 8 + 3 + 8 + 9 + 2 + 6 + 1 + 7 + 4 + 4 + 9 + 6 + 6 + 4 + 4 + 2 + 9 + 5 + 3 + 0 + 2 + 0 + 1 + 3 + 4 + 2 + 2 + 8 + 1 + 8 + 7 + 9 + 1 + 9 + 4 + 6 + 3 + 0 + 8 + 7 + 2 + 4 + 8 + 3 + 2 + 2 + 1 + 4 + 7 + 6 + 5 + 1.

Robert G. Wilson v, Table of n, a(n) for n = 1..546

Cf. A048997, A238104, A051003, A001913.

fQ[n_] := Total[ RealDigits[ 1/n, 10][[1, 1]]] == 666;  Select[ Range@ 25000, fQ ] (* Robert G. Wilson v, Aug 16 2014 *)

A248504 Least number k > 0 such that n^k contains 666 in its decimal representation, or 0 if no such k exists.

Original entry on oeis.org

0, 157, 34, 96, 102, 18, 70, 64, 17, 0, 42, 41, 25, 44, 30, 48, 16, 97, 30, 157, 50, 33, 15, 35, 51, 12, 35, 10, 34, 34, 34, 44, 44, 30, 47, 9, 20, 46, 23, 96, 33, 13, 42, 32, 39, 17, 8, 27, 35, 102, 22, 42, 80, 55, 28, 55, 38, 19, 48, 18, 74, 15, 31, 32, 37

Offset: 1

Talha Ali, Dec 01 2014

a(n) <= a(2) = 157 for all n <= 10^5. Is there any n for which a(n) > 157? - Robert Israel, Dec 01 2014

			a(2)=157 because 2^157=182687704666362864775460604089535377456991567872 contains '666' (see A007356).
a(3)=34 because 3^34=16677181699666568 contains '666' and belongs to A051003.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A007356, A051003.

f:= proc(n)
local k;
if n = 10^ilog10(n) then return 0 fi;
for k from 1 do
  if StringTools[Search]("666",sprintf("%d",n^k)) <> 0 then return k fi
od
end proc;
seq(f(n), n=1..1000); # Robert Israel, Dec 01 2014

A248504[n_] := If[IntegerQ[Log10[n]], 0, Block[{k = 0}, While[StringFreeQ[IntegerString[n^++k], "666"]]; k]];
Array[A248504, 100] (* Paolo Xausa, Apr 08 2024 *)

isok(n) = {d = digits(n); for (i=1, #d-3, if ((d[i] == 6) && (d[i+1]==6) && (d[i+2]==6), return(1));); return (0);}
a(n) = {if ((n==1) || (n==10) || (ispower(n,,&p) && (p==10)), return(0)); k = 1; while (! isok(n^k), k++); k;} \\ Michel Marcus, Dec 01 2014

More terms from Alois P. Heinz, Dec 01 2014

A260312 Palindromic beastly primes that begin and end with digit '1'.

Original entry on oeis.org

16661, 1000000000000066600000000000001, 10000000000000000000000000000000000000000006660000000000000000000000000000000000000000001

Offset: 1

K. D. Bajpai, Jul 22 2015

The next term a(4) contains 1017 digits, and is too large to include in data section.

			a(1) = 16661 is a palindromic prime that contains the beastly number '666' and begins and ends with digit 1.
a(2) = 1000000000000066600000000000001 is palindromic prime that contains the beastly number '666' and begins and ends with digit 1.

Tony Padilla and Brady Haran, The Most Evil Number, Numberphile video (2018)

Cf. A046720, A051003, A131645, A186086.

[k: n in [1..1000] | IsPrime(k) where k is ((1*10^(n + 2) + 666)*10^n + 1 )];

A260312:= n-> ((1*10^(n + 2) + 666)*10^n + 1 ): select(isprime, [seq((A260312 (n), n=1..100))]);

Select[Table[(1*10^(n + 2) + 666)*10^n + 1, {n, 1000}], PrimeQ]
Select[Table[FromDigits[Join[{1},PadRight[{},n,0],{6,6,6},PadRight[ {},n,0],{1}]],{n,0,50}],PrimeQ] (* Harvey P. Dale, Jul 09 2017 *)

for(n=1, 500, k=((1*10^(n + 2) + 666)*10^n + 1 ); if(isprime(k), print1(k, ", ")));

A276563 Digits of the Leviathan number (10^666)!.

Original entry on oeis.org

1, 3, 4, 0, 7, 2, 7, 3, 8, 4, 6, 9, 7, 8, 7, 1, 2, 5, 0, 8, 0, 5, 6, 9, 8, 3, 7, 5, 4, 0, 5, 0, 8, 2, 5, 8, 2, 6, 8, 0, 5, 0, 6, 4, 2, 7, 0, 6, 7, 0, 4, 9, 6, 3, 5, 6, 6, 7, 9, 5, 8, 5, 6, 0, 1, 5, 6, 2, 0, 6, 5, 9, 2, 1, 4, 8, 3, 3, 1, 9, 3, 8, 2, 6, 9, 9, 6

Offset: 1

Martin Renner, Nov 16 2016

The factorial of 10^666, called the Leviathan number by Clifford A. Pickover, is 10^(6.655657055...*10^668), which means that it has approximately 6.656*10^668 decimal digits. The number of trailing zeros is Sum_{k=1..952} floor(10^666/5^k) = 25*10^664 - 143. The last nonzero digits are ...708672.

Clifford A. Pickover: Wonders of Numbers. Adventures in Mathematics, Mind, and Meaning. New York: Oxford University Press, 2001, p. 351.

Martin Renner, Table of n, a(n) for n = 1..984
Robert P. Munafo, Notable Properties of Specific Numbers - (10^666)!
Eric W. Weisstein, Leviathan number. From MathWorld - A Wolfram Web Resource.

Cf. A051003.

cp's OEIS Frontend

A002282 a(n) = 8*(10^n - 1)/9.

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

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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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A046720 Subsequence of beastly primes (A186086) that are palindromes that begin and end with 7.

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A138563 Beastly fax numbers: numbers containing the fax number of the Beast (667, one more than its regular number) in their decimal expansion.

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A098783 Smallest integer not occurring earlier such that the concatenation of the integers has the property that the first two digits sum to 9, the next two digits also, etc.

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A244661 Beastly reciprocals, or numbers n such that digitsum(1/n) = 666.

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A248504 Least number k > 0 such that n^k contains 666 in its decimal representation, or 0 if no such k exists.

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