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A093301 Earliest positive integer having embedded exactly k distinct primes.

%I A093301 #25 Apr 23 2024 07:56:10
%S A093301 1,2,13,23,113,137,1131,1137,1373,11379,11317,23719,111317,113171,
%T A093301 211373,1113171,1113173,1317971,2313797,11131733,11317971,13179719,
%U A093301 82337397,52313797,113179719,113733797,523137971,1113173331,1131797193,1797193373,2113733797,11131733311,11719337397
%N A093301 Earliest positive integer having embedded exactly k distinct primes.
%H A093301 Robert G. Wilson v, <a href="/A093301/b093301.txt">Table of n, a(n) for n = 0..38</a>
%H A093301 Carlos Rivera, <a href="http://www.primepuzzles.net/puzzles/puzz_265.htm">Puzzle 265. Primes embedded</a>, The Prime Puzzles & Problems Connection.
%F A093301 A039997(a(n)) = n and A039997(m) <> n for m < a(n). - _Reinhard Zumkeller_, Jul 16 2007
%e A093301 For example: a(5) = 137 because 137 is the earliest number that has embedded 5 distinct primes: 3, 7, 13, 37 & 137.
%t A093301 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 *)
%o A093301 (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)<n*10,if(isprime(n%t),v=concat(v,n%t)));v=vecsort(v,,8);n\=10);v
%o A093301 print1(1);r=0;for(n=1,1e6,t=#dp(n);if(t>r,r=t;print1(", "n))) \\ _Charles R Greathouse IV_, Jul 10 2012
%o A093301 (Python)
%o A093301 from sympy import isprime
%o A093301 from itertools import count, islice
%o A093301 def A039997(n):
%o A093301     s = str(n)
%o A093301     ss = (int(s[i:j]) for i in range(len(s)) for j in range(i+1, len(s)+1))
%o A093301     return len(set(k for k in ss if isprime(k)))
%o A093301 def agen():
%o A093301     adict, n = dict(), 0
%o A093301     for k in count(1):
%o A093301         v = A039997(k)
%o A093301         if v not in adict: adict[v] = k
%o A093301         while n in adict: yield adict[n]; n += 1
%o A093301 print(list(islice(agen(), 14))) # _Michael S. Branicky_, Aug 07 2022
%Y A093301 Cf. A000040, A039997.
%K A093301 base,nonn
%O A093301 0,2
%A A093301 _Carlos Rivera_, Apr 24 2004
%E A093301 Name clarified, offset corrected, and a(9) inserted by _Michael S. Branicky_, Aug 07 2022
%E A093301 a(22) inserted and a(30)-a(38) added by _Robert G. Wilson v_, Apr 10 2024