A038614 Primes not containing the digit '6'.

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 271, 277, 281, 283, 293, 307, 311

Offset: 1

Author

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

Keywords

• nonn
• easy
• base

Comments

Subsequence of primes of A052414. - 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

• Robert Israel, 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).
• Index to entries for primes with digits in a given set

Crossrefs

Intersection of A000040 (primes) and A052414 (numbers with no digit 6).

Primes having no digit d = 0..9 are A038618, A038603, A038604, A038611, A038612, A038613, this sequence, A038615, A038616, and A038617, respectively.

Programs

Magma

[ p: p in PrimesUpTo(400) | not 6 in Intseq(p) ]; // Bruno Berselli, Aug 08 2011

Maple

no6:= proc(n) option remember;
  n mod 10 <> 6 and procname(floor(n/10))
end proc:
no6(0):= true:
select(no6 and isprime, [2,seq(i,i=3..1000,2)]); # Robert Israel, Mar 16 2017

Mathematica

Select[Prime[Range[70]], DigitCount[#, 10, 6] == 0 &] (* Vincenzo Librandi, Aug 08 2011 *)

PARI

lista(nn)=forprime(p=2, nn, if (!vecsearch(vecsort(digits(p),,8), 6), print1(p, ", "));); \\ Michel Marcus, Feb 22 2015

PARI

select( {is_A038614(n)=is_A052414(n)&&isprime(n)}, [1..350]) \\ see A052414
(A038614_upto(n)=select( is_A038614, primes([1,n])))(350) \\ needs the above
next_A038614(n)={until(isprime(n), n=next_A052414(nextprime(n+1)-1));n}
(A038614_vec(n)=vector(n,i,n=next_A038614(if(i>1,n))))(66) \\ M. F. Hasler, Jan 12 2020

Formula

Intersection of A000040 and A052414. - M. F. Hasler, Jan 12 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