A093301 Earliest positive integer having embedded exactly k distinct primes.

1, 2, 13, 23, 113, 137, 1131, 1137, 1373, 11379, 11317, 23719, 111317, 113171, 211373, 1113171, 1113173, 1317971, 2313797, 11131733, 11317971, 13179719, 82337397, 52313797, 113179719, 113733797, 523137971, 1113173331, 1131797193, 1797193373, 2113733797, 11131733311, 11719337397

Offset: 0

Author

Carlos Rivera, Apr 24 2004

Keywords

• base
• nonn

Examples

			For example: a(5) = 137 because 137 is the earliest number that has embedded 5 distinct primes: 3, 7, 13, 37 & 137.
		

Links

• Robert G. Wilson v, Table of n, a(n) for n = 0..38
• Carlos Rivera, Puzzle 265. Primes embedded, The Prime Puzzles & Problems Connection.

Crossrefs

Cf. A000040, A039997.

Programs

Mathematica

f[n_] := Block[{id = IntegerDigits@ n, lst, len}, len = Length@ id; lst = FromDigits@# & /@ Flatten[ Table[ Take[id, {i, j}], {i, 1, len}, {j, i, len}], 1]; Count[ PrimeQ@ Union@ lst, True]] (* after David W. Wilson in A039997 *); t[] := 0; t[1] = 2; k = 1; While[k < 10000000001, a = f@ k; If[ t[a] == 0, t[a] = k; Print[{a, k}]]; k += 2]; t /@ Range[0, 28] (* _Robert G. Wilson v, Apr 10 2024 *)

PARI

dp(n)=if(n<12,return(if(isprime(n),[n],[])));my(v=vecsort(select(isprime, eval(Vec(Str(n)))),,8),t);while(n>9,if(gcd(n%10,10)>1,n\=10;next);t=10; while((t*=10)r,r=t;print1(", "n))) \\ Charles R Greathouse IV, Jul 10 2012

Python

from sympy import isprime
from itertools import count, islice
def A039997(n):
    s = str(n)
    ss = (int(s[i:j]) for i in range(len(s)) for j in range(i+1, len(s)+1))
    return len(set(k for k in ss if isprime(k)))
def agen():
    adict, n = dict(), 0
    for k in count(1):
        v = A039997(k)
        if v not in adict: adict[v] = k
        while n in adict: yield adict[n]; n += 1
print(list(islice(agen(), 14))) # Michael S. Branicky, Aug 07 2022

Formula

A039997(a(n)) = n and A039997(m) <> n for m < a(n). - Reinhard Zumkeller, Jul 16 2007

Extensions

Name clarified, offset corrected, and a(9) inserted by Michael S. Branicky, Aug 07 2022

a(22) inserted and a(30)-a(38) added by Robert G. Wilson v, Apr 10 2024