A069684 -id:A069684 - cp's OEIS frontend

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

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

A069676 Primes with either no internal digits or all internal digits are 1.

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, 113, 211, 311, 313, 317, 419, 613, 617, 619, 719, 811, 911, 919, 1117, 2111, 2113, 3119, 4111, 5113, 5119, 6113, 8111, 8117, 11113, 11117, 11119, 41113, 41117, 71119

Offset: 1

Amarnath Murthy, Apr 06 2002

Eder Vanzei, Table of n, a(n) for n = 1..257

Cf. A069675, A069677, A069678, A069679, A069680, A069681, A069682, A069683, A069684.

aid1Q[n_]:=Union[Most[Rest[IntegerDigits[n]]]]=={1}; Join[Prime[Range[PrimePi[ 100]]], Select[Prime[Range[PrimePi[100]+1,7100]],aid1Q]] (* Harvey P. Dale, Apr 12 2013 *)

lista(na) = my(vp = List(primes(primepi(100)))); for (n=1, na, my(x=(10^n-1)/9); for (i=1, 9, forstep(j=1, 9, 2, my(y=10^(n+1)*i + 10*x + j); if (ispseudoprime(y), listput(vp, y));););); Vec(vp); \\ Michel Marcus, Jul 12 2022

Corrected by Ray Chandler, Nov 24 2003

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

A069677 Primes with either no internal digits or all internal digits are 2.

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, 127, 223, 227, 229, 421, 521, 523, 727, 821, 823, 827, 829, 929, 1223, 1229, 2221, 3221, 3229, 4229, 5227, 6221, 6229, 7229, 8221, 9221, 9227, 12227, 22229, 42221

Offset: 1

Amarnath Murthy, Apr 06 2002

Michael S. Branicky, Table of n, a(n) for n = 1..275 (1..80 from Harvey P. Dale, 81..178 from David A. Corneth, all terms with <= 1000 digits)

Cf. A069675, A069676, A069678, A069679, A069680, A069681, A069682, A069683, A069684.

Join[Prime[Range[25]],Select[Prime[Range[26,4500]],Union[Most[ Rest[ IntegerDigits[ #]]]] =={2}&]] (* Harvey P. Dale, Aug 12 2021 *)

uptoqdigits(n) = { my(ld = [1,3,7,9]); n = max(n, 2); res = List(primes(primepi(97))); for(i = 1, n-2, twos = 20*(10^i\9); for(j = 1, 9, for(k = 1, #ld, c = j*10^(i+1) + twos + ld[k]; if(isprime(c), listput(res, c) ) ) ) ); Set(res) } \\ David A. Corneth, Aug 12 2021

from sympy import isprime
def agen(maxdigits):
    yield from [2, 3, 5, 7]
    for d in range(2, maxdigits+1):
        pow10, mid = 10**(d-1), 0 if d < 3 else 10*int('2'*(d-2))
        cands = (a*pow10+mid+b for a in range(1, 10) for b in [1, 3, 7, 9])
        yield from filter(isprime, cands)
print([an for an in agen(100)]) # Michael S. Branicky, Aug 12 2021

Corrected by Ray Chandler, Nov 24 2003

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

A069678 Primes with either no internal digits or all internal digits are 3.

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, 131, 137, 139, 233, 239, 331, 337, 431, 433, 439, 631, 733, 739, 839, 937, 2333, 2339, 3331, 4337, 4339, 5333, 6337, 7331, 7333, 9337, 13331, 13337, 13339, 23333, 23339

Offset: 1

Amarnath Murthy, Apr 06 2002

Harvey P. Dale, Table of n, a(n) for n = 1..267

Cf. A069675, A069676, A069677, A069679, A069680, A069681, A069682, A069683, A069684.

Join[Prime[Range[25]],Select[Flatten[Table[10FromDigits[PadRight[{n},k,3]]+d,{n,9},{d,{1,3,7,9}},{k,2,5}]],PrimeQ]]//Sort (* Harvey P. Dale, Aug 08 2020 *)

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

A069679 Primes with either no internal digits or all internal digits are 4.

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, 149, 241, 347, 349, 443, 449, 541, 547, 641, 643, 647, 743, 941, 947, 1447, 2441, 2447, 3449, 4441, 4447, 5441, 5443, 5449, 6449, 8443, 8447, 14447, 14449, 24443, 44449

Offset: 1

Amarnath Murthy, Apr 06 2002

Harvey P. Dale, Table of n, a(n) for n = 1..77 (all terms up to the one millionth prime)

Cf. A069675, A069676, A069677, A069678, A069680, A069681, A069682, A069683, A069684.

Join[Prime[Range[25]],Select[Prime[Range[26,5000]],Union[Most[Rest[ IntegerDigits[ #]]]] =={4}&]] (* Harvey P. Dale, Dec 08 2022 *)

Corrected by Ray Chandler, Nov 24 2003

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

A069680 Primes with either no internal digits or all internal digits are 5.

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, 151, 157, 251, 257, 353, 359, 457, 557, 653, 659, 751, 757, 853, 857, 859, 953, 1553, 1559, 2551, 2557, 3557, 3559, 5557, 6551, 6553, 7559, 9551, 15551, 15559, 45553

Offset: 1

Amarnath Murthy, Apr 06 2002

Cf. A069675, A069676, A069677, A069678, A069679, A069681, A069682, A069683, A069684.

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

A069681 Primes with either no internal digits or all internal digits are 6.

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, 163, 167, 263, 269, 367, 461, 463, 467, 563, 569, 661, 761, 769, 863, 967, 1663, 1667, 1669, 2663, 4663, 5669, 6661, 7669, 8663, 8669, 9661, 16661, 26669, 46663, 56663

Offset: 1

Amarnath Murthy, Apr 06 2002

Cf. A069675, A069676, A069677, A069678, A069679, A069680, A069682, A069683, A069684.

Corrected by Ray Chandler, Nov 24 2003

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

A069682 Primes with either no internal digits or all internal digits are 7.

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, 173, 179, 271, 277, 373, 379, 479, 571, 577, 673, 677, 773, 877, 971, 977, 1777, 2777, 3779, 5779, 6779, 8779, 27773, 27779, 47777, 47779, 57773, 67777, 77773, 97771

Offset: 1

Amarnath Murthy, Apr 06 2002

Cf. A069675, A069676, A069677, A069678, A069679, A069680, A069681, A069683, A069684.

Corrected by Ray Chandler, Nov 24 2003

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

A069683 Primes with either no internal digits or all internal digits are 8.

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, 181, 281, 283, 383, 389, 487, 587, 683, 787, 881, 883, 887, 983, 1889, 2887, 3881, 3889, 4889, 5881, 6883, 7883, 8887, 9883, 9887, 48883, 48889, 58889, 68881, 78887

Offset: 1

Amarnath Murthy, Apr 06 2002

Cf. A069675, A069676, A069677, A069678, A069679, A069680, A069681, A069682, A069684.

Corrected by Ray Chandler, Nov 24 2003

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

A069693 Triangular numbers with either no internal digits or all internal digits are 0.

Original entry on oeis.org

0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 300, 406, 703, 903, 3003, 4005, 8001

Offset: 1

Amarnath Murthy, Apr 06 2002

			The internal digits of 3003 are "00", which are both 0. 15 has no internal digits.

Max Alekseyev, Proof of finiteness of A069693 - A069700

Cf. A069675-A069684, A069693-A069700.

Do[ If[ Union[ Drop[ RotateLeft[ IntegerDigits[n(n + 1)/2]], -2]] == {0}, Print[n(n + 1)/2]], {n, 14, 2 10^6}]

Corrected by Sascha Kurz, Jan 02 2003

cp's OEIS Frontend

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

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A069676 Primes with either no internal digits or all internal digits are 1.

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A069677 Primes with either no internal digits or all internal digits are 2.

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A069678 Primes with either no internal digits or all internal digits are 3.

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A069679 Primes with either no internal digits or all internal digits are 4.

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A069680 Primes with either no internal digits or all internal digits are 5.

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A069681 Primes with either no internal digits or all internal digits are 6.

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A069682 Primes with either no internal digits or all internal digits are 7.

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A069683 Primes with either no internal digits or all internal digits are 8.

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A069693 Triangular numbers with either no internal digits or all internal digits are 0.