A038611 Primes not containing the digit '3'.

2, 5, 7, 11, 17, 19, 29, 41, 47, 59, 61, 67, 71, 79, 89, 97, 101, 107, 109, 127, 149, 151, 157, 167, 179, 181, 191, 197, 199, 211, 227, 229, 241, 251, 257, 269, 271, 277, 281, 401, 409, 419, 421, 449, 457, 461, 467, 479, 487, 491, 499, 509, 521, 541, 547, 557

Offset: 1

Author

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

Keywords

• nonn
• easy
• base

Comments

Subsequence of primes of A052405. - Michel Marcus, Feb 22 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

Intersection of A000040 (primes) and A052405 (numbers with no digit 3).

Primes having no digit d = 0..9 are A038618, A038603, A038604, this sequence, A038612, A038613, A038614, A038615, A038616, and A038617, respectively.

Programs

Magma

[ p: p in PrimesUpTo(600) | not 3 in Intseq(p) ]; // Bruno Berselli, Aug 08 2011

Mathematica

Select[Prime[Range[70]], DigitCount[#, 10, 3] == 0 &] (* Vincenzo Librandi, Aug 08 2011 *)

PARI

lista(nn)=forprime(p=2, nn, if (!vecsearch(vecsort(digits(p),,8), 3), print1(p, ", "));); \\ Michel Marcus, Feb 22 2015

PARI

( {A038611_upto(N,M=1)=select( is_A052405, primes([M,N]))} )(350)

PARI

next_A038611(n)={until((n=nextprime(n+1))==n=next_A052405(n-1),);n}
( {A038611_vec(n,M=2)=M--;vector(n,i,M=next_A038611(M))} )(20, 1000)
\\ Get 20 terms >= 1000. See also OEIS wiki page. - M. F. Hasler, Jan 14 2020

Python

from sympy import isprime
i=j=1
while j<=5000:
    if isprime(i) and "3" not in str(i):
        print(str(j)+" "+str(i))
        j+=1
    i+=1 # Indranil Ghosh, Feb 07 2017

Formula

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