A038604 Primes not containing the digit '2'.

3, 5, 7, 11, 13, 17, 19, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397

Offset: 1

Author

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

Keywords

• nonn
• easy
• base

Comments

Subsequence of primes of A052404. - Michel Marcus, Feb 21 2015

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
• 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

Cf. A000040, A052404.

Subsequence of A065091 (odd primes).

Primes having no digit d = 0..9 are A038618, A038603, this sequence, A038611, A038612, A038613, A038614, A038615, A038616, and A038617, respectively.

Programs

Magma

[ p: p in PrimesUpTo(400) | not 2 in Intseq(p) ]; // Bruno Berselli, Aug 08 2011

Mathematica

Select[Prime[Range[70]], DigitCount[#, 10, 2] == 0 &] (* Vincenzo Librandi, Aug 08 2011 *)

PARI

lista(nn, d=2) = {forprime(p=2, nn, if (!vecsearch(vecsort(digits(p),,8), d), print1(p, ", ")););} \\ Michel Marcus, Feb 21 2015

PARI

select( {is_A038604(n)=is_A052404(n)&&isprime(n)}, [1..400]) \\ see Wiki for more
{next_A038604(n)=until((n==nextprime(n+1))==n=next_A052404(n-1),);n} \\ M. F. Hasler, Jan 12 2020

Python

from sympy import isprime, nextprime
def is_A038604(n): return str(n).find('2')<0 and isprime(n)
def next_A038604(n): # get smallest term > n
    while True:
        n = nextprime(n); s = str(n); t = s.find('2')
        if t < 0: return n
        t = 10**(len(s)-1-t); n += t - n%t
def A038604_upto(stop=math.inf, start=3):
    while start < stop: yield start; start = next_A038604(start)
list(A038604_upto(400))
# M. F. Hasler, Jan 12 2020

Formula

Intersection of A000040 and A052404. - M. F. Hasler, Jan 11 2020

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

Extensions

Offset corrected by Arkadiusz Wesolowski, Aug 07 2011