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A038604 Primes not containing the digit '2'.

%I A038604 #81 Jul 23 2025 00:24:48
%S A038604 3,5,7,11,13,17,19,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101,
%T A038604 103,107,109,113,131,137,139,149,151,157,163,167,173,179,181,191,193,
%U A038604 197,199,307,311,313,317,331,337,347,349,353,359,367,373,379,383,389,397
%N A038604 Primes not containing the digit '2'.
%C A038604 Subsequence of primes of A052404. - _Michel Marcus_, Feb 21 2015
%C A038604 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
%H A038604 Indranil Ghosh, <a href="/A038604/b038604.txt">Table of n, a(n) for n = 1..50000</a>
%H A038604 M. F. Hasler, <a href="/wiki/Numbers_avoiding_certain_digits">Numbers avoiding certain digits</a> OEIS wiki, Jan 12 2020.
%H A038604 James Maynard, <a href="http://arxiv.org/abs/1604.01041">Primes with restricted digits</a>, arXiv:1604.01041 [math.NT], 2016.
%H A038604 James Maynard and Brady Haran, <a href="https://www.youtube.com/watch?v=eeoBCS7IEqs">Primes without a 7</a>, Numberphile video (2019).
%H A038604 <a href="/index/Pri#PrimesWithDigits">Index to entries for primes with digits in a given set</a>
%F A038604 Intersection of A000040 and A052404. - _M. F. Hasler_, Jan 11 2020
%F A038604 a(n) ≍ n^(log 10/log 9) log n. - _Charles R Greathouse IV_, Aug 03 2023
%t A038604 Select[Prime[Range[70]], DigitCount[#, 10, 2] == 0 &] (* _Vincenzo Librandi_, Aug 08 2011 *)
%o A038604 (Magma) [ p: p in PrimesUpTo(400) | not 2 in Intseq(p) ]; // _Bruno Berselli_, Aug 08 2011
%o A038604 (PARI) lista(nn, d=2) = {forprime(p=2, nn, if (!vecsearch(vecsort(digits(p),,8), d), print1(p, ", ")););} \\ _Michel Marcus_, Feb 21 2015
%o A038604 (PARI)
%o A038604 select( {is_A038604(n)=is_A052404(n)&&isprime(n)}, [1..400]) \\ see Wiki for more
%o A038604 {next_A038604(n)=until((n==nextprime(n+1))==n=next_A052404(n-1),);n} \\ _M. F. Hasler_, Jan 12 2020
%o A038604 (Python)
%o A038604 from sympy import isprime, nextprime
%o A038604 def is_A038604(n): return str(n).find('2')<0 and isprime(n)
%o A038604 def next_A038604(n): # get smallest term > n
%o A038604     while True:
%o A038604         n = nextprime(n); s = str(n); t = s.find('2')
%o A038604         if t < 0: return n
%o A038604         t = 10**(len(s)-1-t); n += t - n%t
%o A038604 def A038604_upto(stop=math.inf, start=3):
%o A038604     while start < stop: yield start; start = next_A038604(start)
%o A038604 list(A038604_upto(400))
%o A038604 # _M. F. Hasler_, Jan 12 2020
%Y A038604 Cf. A000040, A052404.
%Y A038604 Subsequence of A065091 (odd primes).
%Y A038604 Primes having no digit d = 0..9 are A038618, A038603, this sequence, A038611, A038612, A038613, A038614, A038615, A038616, and A038617, respectively.
%K A038604 nonn,easy,base
%O A038604 1,1
%A A038604 Vasiliy Danilov (danilovv(AT)usa.net), Jul 15 1998
%E A038604 Offset corrected by _Arkadiusz Wesolowski_, Aug 07 2011