A038618 - OEIS

A038618

Primes not containing the decimal digit 0, a.k.a. zeroless or zerofree primes.

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

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

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

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

MATHEMATICA

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

PROG

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

(Haskell)

a038618 n = a038618_list !! (n-1)

a038618_list = filter ((== 1) . a168046) a000040_list

-- Reinhard Zumkeller, Apr 07 2014, Sep 27 2011

(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

CROSSREFS

Cf. A010051, A168046.

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.