A038603 Primes not containing the digit '1'.

2, 3, 5, 7, 23, 29, 37, 43, 47, 53, 59, 67, 73, 79, 83, 89, 97, 223, 227, 229, 233, 239, 257, 263, 269, 277, 283, 293, 307, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 409, 433, 439, 443, 449, 457, 463, 467, 479, 487, 499, 503, 509, 523, 547, 557

Offset: 1

Author

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

Keywords

• nonn
• easy
• base

Comments

Subsequence of A132080. - Reinhard Zumkeller, Aug 09 2007

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

• Indranil Ghosh, Table of n, a(n) for n = 1..50000 (terms 1..1000 from R. Zumkeller)
• 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).
• Index to entries for primes with digits in a given set

Crossrefs

Intersection of A000040 (primes) and A052383 (numbers with no digit 1).

Primes having no digit d = 0..9 are A038618, this sequence, A038604, A038611, A038612, A038613, A038614, A038615, A038616, and A038617, respectively.

Primes with other restrictions on digits: A106116, A156756.

Programs

Magma

[ p: p in PrimesUpTo(600) | not 1 in Intseq(p) ];  // Bruno Berselli, Aug 08 2011

Mathematica

Select[Prime[Range[70]], DigitCount[#, 10, 1] == 0 &] (* Vincenzo Librandi, Aug 09 2011 *)

PARI

is(n)=if(isprime(n),n=vecsort(eval(Vec(Str(n))),,8);n[1]>1||(!n[1]&&n[2]>1)) \\ Charles R Greathouse IV, Aug 09 2011

PARI

is(n)=!vecsearch(vecsort(digits(n)),1) && isprime(n) \\ Charles R Greathouse IV, Oct 03 2012

PARI

next_A038603(n)=until((n=nextprime(n+1))==n=next_A052383(n-1),);n \\ Compute least a(k) > n. See A052383. - M. F. Hasler, Jan 14 2020

Python

from sympy import nextprime
i=p=1
while i<=500:
    p = nextprime(p)
    if '1' not in str(p):
        print(str(i)+" "+str(p))
        i+=1
# Indranil Ghosh, Feb 07 2017, edited by M. F. Hasler, Jan 15 2020
# See the OEIS Wiki page for more efficient programs. - M. F. Hasler, Jan 14 2020

Formula

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