A132080 -id:A132080 - cp's OEIS frontend

A038603 Primes not containing the digit '1'.

2, 3, 5, 7, 23, 29, 37, 43, 47, 53, 59, 67, 73, 79, 83, 89, 97, 223, 227, 229, 233, 239, 257, 263, 269, 277, 283, 293, 307, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 409, 433, 439, 443, 449, 457, 463, 467, 479, 487, 499, 503, 509, 523, 547, 557

Offset: 1

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

Subsequence of A132080. - Reinhard Zumkeller, Aug 09 2007

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

Indranil Ghosh, Table of n, a(n) for n = 1..50000 (terms 1..1000 from R. Zumkeller)
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

Intersection of A000040 (primes) and A052383 (numbers with no digit 1).
Primes having no digit d = 0..9 are A038618, this sequence, A038604, A038611, A038612, A038613, A038614, A038615, A038616, and A038617, respectively.
Primes with other restrictions on digits: A106116, A156756.

[ p: p in PrimesUpTo(600) | not 1 in Intseq(p) ];  // Bruno Berselli, Aug 08 2011

Select[Prime[Range[70]], DigitCount[#, 10, 1] == 0 &] (* Vincenzo Librandi, Aug 09 2011 *)

is(n)=if(isprime(n),n=vecsort(eval(Vec(Str(n))),,8);n[1]>1||(!n[1]&&n[2]>1)) \\ Charles R Greathouse IV, Aug 09 2011

is(n)=!vecsearch(vecsort(digits(n)),1) && isprime(n) \\ Charles R Greathouse IV, Oct 03 2012

next_A038603(n)=until((n=nextprime(n+1))==n=next_A052383(n-1),);n \\ Compute least a(k) > n. See A052383. - M. F. Hasler, Jan 14 2020

from sympy import nextprime
i=p=1
while i<=500:
    p = nextprime(p)
    if '1' not in str(p):
        print(str(i)+" "+str(p))
        i+=1
# Indranil Ghosh, Feb 07 2017, edited by M. F. Hasler, Jan 15 2020
# See the OEIS Wiki page for more efficient programs. - M. F. Hasler, Jan 14 2020

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

A316227 Composite numbers k for which no nontrivial divisor shares any digits with k.

Original entry on oeis.org

4, 6, 8, 9, 10, 14, 16, 18, 21, 27, 34, 38, 46, 49, 54, 56, 57, 58, 68, 69, 76, 78, 81, 86, 87, 106, 111, 116, 118, 129, 134, 146, 158, 161, 166, 177, 188, 201, 219, 247, 249, 259, 267, 289, 323, 329, 334, 356, 358, 388, 413, 446, 454, 458, 466, 477, 478, 489

Offset: 1

Jason Zimba, Jun 27 2018

A nontrivial divisor of k means a divisor greater than 1 and less than k.

			The nontrivial divisors of 54 are 2, 3, 6, 9, 18, and 27, none of which have a digit 5 or 4.
The nontrivial divisors of 248629501 are 337 and 737773.
The nontrivial divisors of 321810649 are 557 and 577757.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A002808, A035139, A132080.

filter:= proc(n) local S;
  if isprime(n) then return false fi;
  S:= convert(convert(n,base,10),set);
  andmap(d -> convert(convert(d,base,10),set) intersect S = {}, numtheory:-divisors(n) minus {1,n})
end proc:
select(filter, [$4..1000]); # Robert Israel, Jul 22 2018

MaxCheck = 1000; (* modify as desired *)
ResultList = {};
Do[
  If[
   Not[PrimeQ[k]] &&
    Intersection[
      Flatten[
       Map[
        IntegerDigits,
        Complement[Divisors[k], {1, k}]
        ]
       ],
      IntegerDigits[k]
      ] == {},
   ResultList = Append[ResultList, k]
   ],
  {k, 2, MaxCheck}];
ResultList
(* or *) Select[Range@500, CompositeQ@# && Intersection[ IntegerDigits@#, Flatten@ IntegerDigits@ Most@ Rest@ Divisors@ #] == {} &] (* Giovanni Resta, Jun 27 2018 *)

isok(n) = {my(d=divisors(n), dd = Set(digits(n))); for (k=2, #d-1, if (#setintersect(Set(digits(d[k])), dd), return (0));); return (1);}
lista(nn) = {forcomposite(n=1, nn, if (isok(n), print1(n, ", ")););} \\ Michel Marcus, Jul 03 2018

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A038603 Primes not containing the digit '1'.

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A316227 Composite numbers k for which no nontrivial divisor shares any digits with k.

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