A038618 Primes not containing the digit '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, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293

Offset: 1

Author

Vasiliy Danilov (danilovv(AT)usa.net), Jul 15 1998

Keywords

• nonn
• easy
• base

Comments

Complement of A056709 with respect to primes (A000040). - Lekraj Beedassy, Jul 04 2010

Maynard proves that this sequence is infinite and in particular contains the expected number of elements up to x, on the order of x^(log 9/log 10)/log x. - Charles R Greathouse IV, Apr 08 2016

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• René-Louis Clerc, Nombres premiers primaires et nombres premiers secondaires , 2025.
• M. F. Hasler, Numbers avoiding certain digits, OEIS Wiki, Jan 12 2020.
• James Maynard, Primes with restricted digits, arXiv:1604.01041 [math.NT], 2016.
• James Maynard and Brady Haran, Primes without a 7, Numberphile video (2019).
• Eric Weisstein's World of Mathematics, Zerofree.
• Index to entries for primes with digits in a given set

Crossrefs

Cf. A010051, A168046, A384793, A384794.

Subsequence of A000040 (primes), A052382 (zeroless numbers) and A195943.

Primes having no digit d = 0..9 are this sequence, A038603, A038604, A038611, A038612, A038613, A038614, A038615, A038616, and A038617, respectively.

Programs

Haskell

a038618 n = a038618_list !! (n-1)
a038618_list = filter ((== 1) . a168046) a000040_list
-- Reinhard Zumkeller, Apr 07 2014, Sep 27 2011

Magma

[ p: p in PrimesUpTo(300) | not 0 in Intseq(p) ];  // Bruno Berselli, Aug 08 2011

Mathematica

Select[Prime[Range[70]], DigitCount[#, 10, 0] == 0 &] (* Vincenzo Librandi, Aug 09 2011 *)

PARI

is(n)=if(isprime(n),n=vecsort(eval(Vec(Str(n))),,8);n[1]>0) \\ Charles R Greathouse IV, Aug 09 2011

PARI

lista(nn) = forprime (p=2, nn, if (vecmin(digits(p)), print1(p, ", "))); \\ Michel Marcus, Apr 06 2016

PARI

next_A038618(n)=until(vecmin(digits(n=nextprime(next_A052382(n)))),);n \\ Cf. OEIS Wiki page (LINKS) for other programs. - M. F. Hasler, Jan 12 2020

Python

from sympy import primerange
def aupto(N): return [p for p in primerange(1, N+1) if '0' not in str(p)]
print(aupto(300)) # Michael S. Branicky, Mar 11 2022

Formula

Intersection of A052382 (zeroless numbers) and A000040 (primes); A168046(a(n))*A010051(a(n)) = 1. - Reinhard Zumkeller, Dec 01 2009

a(n) ≍ n^(log 10/log 9) log n. - Charles R Greathouse IV, Aug 03 2023