A164968 -id:A164968 - cp's OEIS frontend

A173029 Partial sums of naughty primes A164968.

10007, 20016, 60025, 130026, 200029, 270038, 360039, 450046, 550049, 750052, 950061, 1250068, 1650077, 2150086, 2850087, 3750088, 4650095, 5650098, 6650131, 7650168, 8650207, 9650288, 10650387, 11650690, 12651093, 13651502, 14652009, 15652618, 16653525

Offset: 1

Jonathan Vos Post, Feb 07 2010

The subsequence of prime partial sums of naughty primes begins: 10007, 200029, 550049, 6650131. The subsubsequence of naughty prime partial sums of naughty primes begins: 10007, and then what? The smallest square in the sequence is 60025 = 5^2 * 7^4.

			a(24) = 10007 + 10009 + 40009 + 70001 + 70003 + 70009 + 90001 + 90007 + 100003 + 200003 + 200009 + 300007 + 400009 + 500009 + 700001 + 900001 + 900007 + 1000003 + 1000033 + 1000037 + 1000039 + 1000081 + 1000099 + 1000303.

Cf. A000040, A164968.

Accumulate[Select[Prime[Range[100000]],DigitCount[#,10,0]> IntegerLength[ #]/2&]] (* Harvey P. Dale, Jun 09 2015 *)

a(n) = SUM[i=1..n] {p such that p is prime and the number of zeros in the decimal representation of p is greater than the number of all other digits}.

Corrected and extended by Harvey P. Dale, Jun 09 2015

A069675 Primes all of whose internal digits (if any) are 0.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 307, 401, 409, 503, 509, 601, 607, 701, 709, 809, 907, 1009, 2003, 3001, 4001, 4003, 4007, 5003, 5009, 6007, 7001, 8009, 9001, 9007, 10007, 10009

Offset: 1

Amarnath Murthy, Apr 06 2002

Despite their initial density, these primes are rare. The value of a(310) = 9*10^2914 + 7. Beginning with a(54), this is a subsequence of A164968. Indeed, these could be called the "naughtiest" primes. - Harlan J. Brothers, Aug 17 2015

There are expected to be infinitely many terms, but growing very rapidly, something like a(n) ~ exp(exp(const * n)). - Robert Israel, Aug 17 2015

			4001 is in the sequence because it is prime and all the internal digits (the digits between 4 and 1) are zero. - _Michael B. Porter_, Aug 11 2016

Charles R Greathouse IV, Table of n, a(n) for n = 1..263
Makoto Kamada, Prime numbers of the form k*10^n+1
Seth A. Troisi, Plot of log(log(a(n))) for 1 <= n <= 434
Seth A. Troisi, a(n) for n = 1 .. 434, a(n) in form "a * 10 ^ d + b"

Cf. A069676-A069684, A164968.

A := {}:
for n to 1000 do
  p := ithprime(n):
  d := convert(p, base, 10):
  s := 0:
  for m from 2 to nops(d)-1 do
    s := s+d[m]:
  end do
  if s = 0 then
    A := `union`(A, {p})
  end if:
end do:
A := A
# César Eliud Lozada, Sep 04 2012
select(isprime, [$1..9, seq(seq(seq(10^d*a+b, b=1..9),a=1..9), d=1..10)]); # Robert Israel, Aug 18 2015

Select[Prime[Range[1, 100000]], IntegerLength[#] < 3 || Union@Rest@Most@IntegerDigits[#, 10] == {0} &] (* Harlan J. Brothers, Aug 17 2015 *)
Select[Join[Range[1, 99], Flatten[Table[a*10^d + b, {d, 2, 50}, {a, 1, 9}, {b, 1, 9}]]], PrimeQ[#] &] (* Seth A. Troisi, Aug 03 2016 *)

go(n)=my(v=List(primes(4)),t); for(d=1,n-1, for(i=1,9, forstep(j=1,9,[2,4,2], if(isprime(t=10^d*i+j), listput(v,t))))); Vec(v) \\ Charles R Greathouse IV, Sep 14 2015

from sympy import isprime
print([2, 3, 5, 7] + list(filter(isprime, (a*10**d+b for d in range(1, 101) for a in range(1, 10) for b in [1, 3, 7, 9])))) # Michael S. Branicky, May 08 2021

a(n) >> 10^(n/24). - Charles R Greathouse IV, Sep 14 2015

Offset corrected and name changed by Arkadiusz Wesolowski, Sep 07 2011

A056709 Naught-y primes, primes with noughts (or zeros).

Original entry on oeis.org

101, 103, 107, 109, 307, 401, 409, 503, 509, 601, 607, 701, 709, 809, 907, 1009, 1013, 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1087, 1091, 1093, 1097, 1103, 1109, 1201, 1301, 1303, 1307, 1409, 1601, 1607, 1609, 1709, 1801, 1901, 1907

Offset: 1

Robert G. Wilson v, Aug 10 2000

Intersection of A000040 and A011540. - Michel Marcus, Mar 12 2015

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Chris Caldwell, Naughty prime, Prime Pages' Glossary (UTM). (Date?)

Cf. A000040 (primes), A011540 (numbers that contain a 0).
Complement, in A000040, of zeroless primes A038618.
Cf. A164968 (Naughty primes: most digits are 0).

[p:p in PrimesUpTo(2000)|0 in Intseq(p)]; // Marius A. Burtea, Jan 13 2020

Select[ Range[ 1, 2500, 2 ], PrimeQ[ # ] && Sort[ RealDigits[ # ][ [ 1 ] ] ][ [ 1 ] ] == 0 & ]
(* Second program: *)
Select[Prime@ Range@ 300, DigitCount[#, 10, 0] > 0 &] (* Michael De Vlieger, Jan 28 2020 *)

is(n)=isprime(n)&&vecsort(eval(Vec(Str(n))),,8)[1]==0

select( {is_A056709(n)=!vecmin(digits(n))&&isprime(n)}, [1..2000]) \\ Defines the characteristic function is_A; as check & example: select terms in [1..2000].
next_A056709(n)={until(!vecmin(digits(n)), n=nextprime(n+1));n} \\ Successor function: find smallest a(k) > n. Useful to get a vector of consecutive terms:
A056709_vec(n,M=99)=M--;vector(n,i,M=next_A056709(M)) \\ get n terms >= M (if given, else start with a(1)).  \\ M. F. Hasler, Jan 12 2020

from sympy import primerange
def aupto(lim): return [p for p in primerange(1, lim+1) if '0' in str(p)]
print(aupto(1910)) # Michael S. Branicky, Mar 11 2022

a(n) ~ n log n: almost all primes are in this sequence. - Charles R Greathouse IV, Jul 24 2012

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

A182051 Primes with a majority of one digit.

Original entry on oeis.org

2, 3, 5, 7, 11, 101, 113, 131, 151, 181, 191, 199, 211, 223, 227, 229, 233, 277, 311, 313, 331, 337, 353, 373, 383, 433, 443, 449, 499, 557, 577, 599, 661, 677, 727, 733, 757, 773, 787, 797, 811, 877, 881, 883, 887, 911, 919, 929, 977, 991, 997, 1117, 1151

Offset: 1

Arkadiusz Wesolowski, Apr 08 2012

a(n+5) = A164937(n) for n <= 89.

			1151 is prime and the number of ones is greater than the number of all other digits, so this number is in the sequence.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..10000

Supersequence of A164937 and of A164968.

Cf. A004022, A105992.

lst = {}; Do[i = IntegerDigits[n]; If[PrimeQ[n] && Count[i, First[Commonest@i]] > IntegerLength[n]/2, AppendTo[lst, n]], {n, 10^4}]; lst

A386247 Primes containing 000 as a substring.

Original entry on oeis.org

10007, 10009, 40009, 70001, 70003, 70009, 90001, 90007, 100003, 100019, 100043, 100049, 100057, 100069, 130003, 140009, 150001, 160001, 160009, 170003, 180001, 180007, 200003, 200009, 200017, 200023, 200029, 200033, 200041, 200063, 200087, 220009, 230003, 240007

Offset: 1

Alois P. Heinz, Jul 16 2025

Differs from A164968 first at n=10: a(10) = 100019 < 200003 = A164968(10).

Alois P. Heinz, Table of n, a(n) for n = 1..20761

Cf. A000040, A164968, A243527, A166580, A166581, A166582, A167281, A131645, A167282, A167290, A167292, A386240.

Select[Prime[Range[1230, 25000]], StringContainsQ[IntegerString[#], "000"] &] (* Paolo Xausa, Jul 19 2025 *)

A176781 Smallest prime prime(i) such that concatenation 2//0_(n)//prime(i) is prime.

Original entry on oeis.org

3, 11, 3, 17, 3, 3, 3, 11, 89, 41, 257, 3, 29, 131, 353, 3, 3, 11, 89, 521, 257, 3, 17, 3, 467, 89, 149, 17, 71, 47, 293, 17, 191, 47, 3, 41, 23, 11, 401, 41, 443, 41, 293, 479, 311, 23, 587, 41, 1289, 1013, 29, 41, 59, 293, 1031, 17, 23, 17, 347, 401, 599, 11, 227, 827, 401

Offset: 0

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Apr 26 2010

We search for the prime such that the first prime (=2) concatenated with n zeros and concatenated with that prime is again a prime number.
If p = prime(i) is a d(i)-digit prime: q = 2 * 10^(n+d(i)) + p has to be prime.
Necessarily prime(i) is congruent to 2 (mod 3).
It is conjectured that prime(i) = 3 occurs infinitely often: at n= 0, 2, 4, 5, 6, 11, 15, 16, 21, 23, 34, 114, 119,...

			n = 0: 2//3 = 23 = prime(9), 3 = prime(2) is first term
n = 1: 2//0//11 = 2011 = prime(305), 11 = prime(5) is 2nd term
n = 2: 2//00//3 = 2003 = prime(304), 3 = prime(2) is 3rd term

E. I. Ignatjew, Mathematische Spielereien, Urania Verlag Leipzig/Jena/ Berlin 1982

Cf. A164968, A173291, A176316

Offset corrected and sequence extended by R. J. Mathar, Apr 28 2010

A216203 Smallest prime that does not divide at least one n-digit zeroless pandigital number.

Original entry on oeis.org

44449, 900001, 7000003, 20000003, 30000001, 100000007, 500000003, 1000000007, 6000000001

Offset: 9

Arkadiusz Wesolowski, Mar 12 2013

How many first terms are in A182051?

The analogous sequence for pandigital numbers is A228253. - Giovanni Resta, Aug 19 2013

Carlos Rivera, Puzzle 259
Eric Weisstein's World of Mathematics, Pandigital Number

Cf. A164968, A182051.

lst = Times @@ Union[FromDigits@# & /@ Permutations@Range[9]]; n = 1; While[True, p = Prime[n]; If[! Divisible[lst, p], Print[p]; Break[]]; n++]

a(11)-a(16) from Giovanni Resta, Mar 12 2013

a(17) from Giovanni Resta, Mar 13 2013

A176833 Smallest prime p = prime(i) such that concatenation q(i) = 13//0_(k)//prime(i) (k = 0, 1, 2, ...) is prime.

Original entry on oeis.org

7, 3, 3, 3, 151, 61, 7, 3, 19, 3, 109, 109, 19, 19, 37, 409, 109, 97, 61, 19, 73, 109, 139, 139, 619, 31, 127, 31, 193, 3, 43, 19, 337, 7, 73, 367, 109, 373, 139, 139

Offset: 1

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Apr 27 2010

See comments in A176781
Necessarily p = 3 or p of form 3 * n + 1
In recreational mathematics some authors call a prime that is composed of mostly naughts, i.e. zeros, a naughty prime

			q(0) = 13//7 = 137 = prime(33), 7 = prime(4) is 1st term
q(1) = 13//0//3 = 1303 = prime(213), 3 = prime(2) is 2nd term
q(26) = 13000000000000000000000000031 is a palindromic prime

A164968, A173291, A176316, A176781

cp's OEIS Frontend

A173029 Partial sums of naughty primes A164968.

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A069675 Primes all of whose internal digits (if any) are 0.

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Maple

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A056709 Naught-y primes, primes with noughts (or zeros).

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Magma

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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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A182051 Primes with a majority of one digit.

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Mathematica

A386247 Primes containing 000 as a substring.

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A176781 Smallest prime prime(i) such that concatenation 2//0_(n)//prime(i) is prime.

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A216203 Smallest prime that does not divide at least one n-digit zeroless pandigital number.