A038617 Primes not containing the digit '9'.

2, 3, 5, 7, 11, 13, 17, 23, 31, 37, 41, 43, 47, 53, 61, 67, 71, 73, 83, 101, 103, 107, 113, 127, 131, 137, 151, 157, 163, 167, 173, 181, 211, 223, 227, 233, 241, 251, 257, 263, 271, 277, 281, 283, 307, 311, 313, 317, 331, 337, 347, 353, 367, 373, 383, 401, 421

Offset: 1

Author

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

Keywords

• nonn
• easy
• base

Comments

Subsequence of primes of A007095. - 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 A007095 (numbers with no digit 9).

Primes having no digit d = 0..9 are A038618, A038603, A038604, A038611, A038612, A038613, A038614, A038615, A038616, and this sequence, respectively.

Primes with other restrictions on digits: A106116, A156756.

Programs

Magma

[ p: p in PrimesUpTo(500) | not 9 in Intseq(p) ]; // Bruno Berselli, Aug 08 2011

Mathematica

Select[Prime[Range[1000]], DigitCount[ # ][[9]] == 0 &] (* Stefan Steinerberger, May 20 2006 *)

PARI

lista(nn)=forprime(p=2, nn, if (!vecsearch(vecsort(digits(p),,8), 9), print1(p, ", "));); \\ Michel Marcus, Feb 22 2015

PARI

lista(nn) = forprime (p=2, nn, if (vecmax(digits(p)) != 9, print1(p, ", "))); \\ Michel Marcus, Apr 06 2016

PARI

next_A038617(n)=until((n=nextprime(n+1))==(n=next_A007095(n-1)), ); n \\ M. F. Hasler, Jan 14 2020

Python

from sympy import isprime
i = 1
while i <= 300:
    if "9" not in str(i) and isprime(i):
        print(str(i), end=",")
    i += 1 # Indranil Ghosh, Feb 07 2017

Formula

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