cp's OEIS Frontend

A038618 Primes not containing the digit '0'.

%I A038618 #85 Jul 23 2025 00:24:20
%S A038618 2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,
%T A038618 97,113,127,131,137,139,149,151,157,163,167,173,179,181,191,193,197,
%U A038618 199,211,223,227,229,233,239,241,251,257,263,269,271,277,281,283,293
%N A038618 Primes not containing the digit '0'.
%C A038618 Complement of A056709 with respect to primes (A000040). - _Lekraj Beedassy_, Jul 04 2010
%C A038618 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 A038618 Charles R Greathouse IV, <a href="/A038618/b038618.txt">Table of n, a(n) for n = 1..10000</a>
%H A038618 René-Louis Clerc, <a href="https://hal.science/hal-05029352">Nombres premiers primaires et nombres premiers secondaires </a>, 2025.
%H A038618 M. F. Hasler, <a href="/wiki/Numbers_avoiding_certain_digits">Numbers avoiding certain digits</a>, OEIS Wiki, Jan 12 2020.
%H A038618 James Maynard, <a href="http://arxiv.org/abs/1604.01041">Primes with restricted digits</a>, arXiv:1604.01041 [math.NT], 2016.
%H A038618 James Maynard and Brady Haran, <a href="https://www.youtube.com/watch?v=eeoBCS7IEqs">Primes without a 7</a>, Numberphile video (2019).
%H A038618 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Zerofree.html">Zerofree</a>.
%H A038618 <a href="/index/Pri#PrimesWithDigits">Index to entries for primes with digits in a given set</a>
%F A038618 Intersection of A052382 (zeroless numbers) and A000040 (primes); A168046(a(n))*A010051(a(n)) = 1. - _Reinhard Zumkeller_, Dec 01 2009
%F A038618 a(n) ≍ n^(log 10/log 9) log n. - _Charles R Greathouse IV_, Aug 03 2023
%t A038618 Select[Prime[Range[70]], DigitCount[#, 10, 0] == 0 &] (* _Vincenzo Librandi_, Aug 09 2011 *)
%o A038618 (Magma) [ p: p in PrimesUpTo(300) | not 0 in Intseq(p) ];  // _Bruno Berselli_, Aug 08 2011
%o A038618 (PARI) is(n)=if(isprime(n),n=vecsort(eval(Vec(Str(n))),,8);n[1]>0) \\ _Charles R Greathouse IV_, Aug 09 2011
%o A038618 (PARI) lista(nn) = forprime (p=2, nn, if (vecmin(digits(p)), print1(p, ", "))); \\ _Michel Marcus_, Apr 06 2016
%o A038618 (PARI) next_A038618(n)=until(vecmin(digits(n=nextprime(next_A052382(n)))),);n \\ Cf. OEIS Wiki page (LINKS) for other programs. - _M. F. Hasler_, Jan 12 2020
%o A038618 (Haskell)
%o A038618 a038618 n = a038618_list !! (n-1)
%o A038618 a038618_list = filter ((== 1) . a168046) a000040_list
%o A038618 -- _Reinhard Zumkeller_, Apr 07 2014, Sep 27 2011
%o A038618 (Python)
%o A038618 from sympy import primerange
%o A038618 def aupto(N): return [p for p in primerange(1, N+1) if '0' not in str(p)]
%o A038618 print(aupto(300)) # _Michael S. Branicky_, Mar 11 2022
%Y A038618 Cf. A010051, A168046, A384793, A384794.
%Y A038618 Subsequence of A000040 (primes), A052382 (zeroless numbers) and A195943.
%Y A038618 Primes having no digit d = 0..9 are this sequence, A038603, A038604, A038611, A038612, A038613, A038614, A038615, A038616, and A038617, respectively.
%K A038618 nonn,easy,base
%O A038618 1,1
%A A038618 Vasiliy Danilov (danilovv(AT)usa.net), Jul 15 1998