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

1,1

Comments

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

Links

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

Example

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

Mathematica

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 *)

Prog

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

Crossrefs

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

Sequence in context: A118985 A092728 A089769 * A071062 A320725 A320771

Adjacent sequences: A033271 A033272 A033273 * A033275 A033276 A033277

Keyword

base,nonn

Author

Michael Kleber

Extensions

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

Status

approved