A186086 -id:A186086 - cp's OEIS frontend

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

76667, 700666007, 700000666000007, 70000006660000007

Offset: 1

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

A051003 Beastly (or hateful) numbers: numbers containing the string 666 in their decimal expansion.

Original entry on oeis.org

666, 1666, 2666, 3666, 4666, 5666, 6660, 6661, 6662, 6663, 6664, 6665, 6666, 6667, 6668, 6669, 7666, 8666, 9666, 10666, 11666, 12666, 13666, 14666, 15666, 16660, 16661, 16662, 16663, 16664, 16665, 16666, 16667, 16668, 16669, 17666, 18666

Offset: 1

Eric W. Weisstein

Jens Kruse Andersen, Table of n, a(n) for n = 1..1000
Tony Padilla and Brady Haran, The Most Evil Number, Numberphile video (2018)
Eric Weisstein's World of Mathematics, Beast Number

Cf. A131645, A057564, A002282, A138563, A043515, A046720, A186086.

Select[Range[18666], ! StringFreeQ[ToString[#], "666"] &] (* Arkadiusz Wesolowski, Sep 08 2011 *)

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

A186521 Numbers n>0 such that (710^(n+2)+666)10^n+7 is prime.

Original entry on oeis.org

1, 3, 6, 7, 49, 578, 793, 1322, 1392, 2392, 3066, 13479

Offset: 1

Arkadiusz Wesolowski, Feb 23 2011

Or, indices of primes in A046720. Rudolf Ondrejka calls these "beastly palindromic primes".

Some of the larger entries may only correspond to probable primes.

Chris Caldwell, The Prime Glossary, Beastly prime
G. L. Honaker, Jr. and Chris Caldwell, Prime Curios! 76667
R. Ondrejka, The Top Ten: a Catalogue of Primal Configurations

Cf. A046720, A186086.

Select[Range[10^3], PrimeQ[(7*10^(# + 2) + 666)*10^# + 7] &] (* Arkadiusz Wesolowski, Sep 08 2011 *)

a(12) added by Arkadiusz Wesolowski, Mar 16 2011

A186538 Numbers n>0 such that 666*10^n+7 is prime.

Original entry on oeis.org

25, 26, 28, 33, 49, 80, 81, 133, 179, 202, 222, 271, 411, 588, 682, 1039, 2403, 5802, 11544, 12484, 41047, 46396

Offset: 1

Arkadiusz Wesolowski, Feb 23 2011

Some of the larger entries may only correspond to probable primes.

Chris Caldwell, The Prime Glossary, Beastly prime
G. L. Honaker, Jr. and Chris Caldwell, Prime Curios! 66600...00007 (28-digits)
Henri & Renaud Lifchitz, PRP Records
R. Ondrejka, The Top Ten: a Catalogue of Primal Configurations

Cf. A186086.

Select[Range[10^3], PrimeQ[666*10^# + 7] &] (* Arkadiusz Wesolowski, Sep 08 2011 *)

a(21) from Arkadiusz Wesolowski, May 25 2011

a(22) from Arkadiusz Wesolowski, May 28 2011

A186630 Numbers n>0 such that 666*10^n+1 is prime.

Original entry on oeis.org

1, 2, 10, 12, 30, 48, 373, 580, 619, 718, 1592, 2928, 3076, 4741, 9198, 14020

Offset: 1

Arkadiusz Wesolowski, Feb 24 2011

No other terms below 10^5. [Arkadiusz Wesolowski, Dec 23 2011]

Chris Caldwell, The Prime Glossary, Beastly prime
G. L. Honaker, Jr. and Chris Caldwell, Prime Curios! 66601
R. Ondrejka, The Top Ten: a Catalogue of Primal Configurations (May 2001), p. 83

Cf. A186086.

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

a(16) (discovered on Dec 09 1999, by Nuutti Kuosa) added by Arkadiusz Wesolowski, Mar 16 2011

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, ", ")));

cp's OEIS Frontend

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

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A051003 Beastly (or hateful) numbers: numbers containing the string 666 in their decimal expansion.

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

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A186521 Numbers n>0 such that (7*10^(n+2)+666)*10^n+7 is prime.

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A186538 Numbers n>0 such that 666*10^n+7 is prime.

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A186630 Numbers n>0 such that 666*10^n+1 is prime.

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

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A186521 Numbers n>0 such that (710^(n+2)+666)10^n+7 is prime.