A089770 -id:A089770 - cp's OEIS frontend

A033274 Primes that do not contain any other prime as a proper substring.

2, 3, 5, 7, 11, 19, 41, 61, 89, 101, 109, 149, 181, 401, 409, 449, 491, 499, 601, 691, 809, 881, 991, 1009, 1049, 1069, 1481, 1609, 1669, 1699, 1801, 4001, 4049, 4481, 4649, 4801, 4909, 4969, 6091, 6469, 6481, 6869, 6949, 8009, 8069, 8081, 8609, 8669, 8681

Offset: 1

Michael Kleber

If there is more than one digit, all digits must be nonprime numbers.
A179335(n) = prime(n) iff prime(n) is in this sequence. For n > 4, prime(n) is in this sequence iff A109066(n) = 0. - Reinhard Zumkeller, Jul 11 2010, corrected by M. F. Hasler, Aug 27 2012
A079066(n) = 0 iff prime(n) is in this sequence. [Corrected by M. F. Hasler, Aug 27 2012]
What are the asymptotics of this sequence? - Charles R Greathouse IV, Aug 27 2012

			149 is a term as 1, 4, 9, 14, 49 are all nonprimes.
199 is not a term as 19 is a prime.

Michael S. Branicky, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Zak Seidov)

Cf. A089768, A089770, A039996, A079397, A033274, A034844, A179909-A179919.

import Data.List (elemIndices)
a033274 n = a033274_list !! (n-1)
a033274_list = map (a000040 . (+ 1)) $ elemIndices 0 a079066_list
-- Reinhard Zumkeller, Jul 19 2011

f[n_] := Block[ {id = IntegerDigits@n}, len = Length@ id - 1; Count[ PrimeQ@ Union[ FromDigits@# & /@ Flatten[ Table[ Partition[ id, k, 1], {k, len}], 1]], True] + 1]; Select[ Prime@ Range@ 1100, f@# == 1 &] (* Robert G. Wilson v, Aug 01 2010 *)
Select[Prime[Range[1100]],NoneTrue[Flatten[Table[FromDigits/@Partition[IntegerDigits[#],d,1],{d,IntegerLength[#]-1}]],PrimeQ]&] (* Harvey P. Dale, Apr 19 2025 *)

from sympy import isprime
def ok(n):
    if n in {2, 3, 5, 7}: return True
    s = str(n)
    if set(s) & {"2", "3", "5", "7"} or not isprime(n): return False
    ss2 = set(s[i:i+l] for i in range(len(s)-1) for l in range(2, len(s)))
    return not any(isprime(int(ss)) for ss in ss2)
print([k for k in range(9000) if ok(k)]) # Michael S. Branicky, Jun 29 2022

Edited by N. J. A. Sloane at the suggestion of Luca Colucci, Apr 03 2008

A089771 Largest n-digit prime containing no prime substrings, or 0 if no such prime exists.

Original entry on oeis.org

7, 89, 991, 9949, 99469, 996649, 9999481, 99990091, 999996901, 9999988049, 99999981469, 999999946849, 9999999946009, 99999999904081, 999999999946069, 9999999999944869, 99999999999884081, 999999999999984469, 9999999999999968869, 99999999999999900049, 999999999999999944009, 9999999999999999999869

Offset: 1

Amarnath Murthy, Nov 23 2003

Conjecture: No term is zero.

Robert Israel, Table of n, a(n) for n = 1..234

Cf. A089768, A033274, A089770.

cbd:= proc(n)
  local x,L,i,flag;
  L:= convert(n,base,10);
  x:= n;
  for i from nops(L) to 1 by -1 do
    if member(L[i],{2,3,5,7}) then
      flag:= true;
      if L[i] = 3 then x:= x - 10^(i-1)- (x mod 10^(i-1)) -1 else x:= x - (x mod 10^(i-1)) - 1 fi;
      return x
    fi;
  od;
 x
end proc:
nps:= proc(n) option remember;
  if n <= 10 then not isprime(n)
  else not isprime(n) and nps(floor(n/10)) and nps(n mod 10^ilog10(n))
  fi
end proc:
npps:= proc(n) local L,i,t;
  if n <= 10 then true
  else
    if n::odd then return nps(floor(n/10)) and nps(n mod 10^ilog10(n)) fi;
    for i from 1 do
      t:= floor(n/10^i);
      if t::odd then return nps(t) and nps(t mod 10^ilog10(t))
      elif t = 0 then return true
  fi od fi
end proc:
g:= proc(n) local p,i,x;
  p:= 10^n;
  do
    p:= prevprime(p);
    x:= cbd(p);
    while x <> p do
      p:= prevprime(x+1);
      x:= cbd(p)
    od;
    if npps(p) then return p fi
  od
end proc:
g(1):= 7:
map(g, [$1..30]); # Robert Israel, Aug 30 2024

from sympy import isprime
from itertools import product
def c(s): return not any(isprime(int(s[i:j])) for i in range(len(s)-1, -1, -1) for j in range(len(s), i, -1) if (i, j) != (0, len(s)))
def a(n):
    if n == 1: return 7
    return next(t for p in product("986410", repeat=n-1) for last in "91" if isprime(t:=int(s:="".join(p)+last)) and c(s))
print([a(n) for n in range(1, 23)]) # Michael S. Branicky, Aug 30 2024

More terms from David Wasserman, Oct 12 2005

More terms from Robert Israel, Aug 30 2024

cp's OEIS Frontend

A033274 Primes that do not contain any other prime as a proper substring.

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