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A346203 a(n) is the smallest nonnegative number k such that the decimal expansion of the product of the first k primes contains the string n.

%I A346203 #17 Jun 15 2022 10:49:14
%S A346203 3,0,1,3,10,7,2,9,9,8,4,18,17,11,15,16,14,18,24,16,11,4,9,5,21,13,13,
%T A346203 13,9,21,3,5,10,14,12,13,26,24,12,17,18,15,12,26,16,22,10,16,12,11,13,
%U A346203 7,13,20,17,19,11,20,15,18,11,14,21,13,10,24,20,14,21,8,9
%N A346203 a(n) is the smallest nonnegative number k such that the decimal expansion of the product of the first k primes contains the string n.
%H A346203 Ilya Gutkovskiy, <a href="/A346203/a346203.jpg">Scatterplot of a(n) up to n=100000</a>
%H A346203 <a href="/index/Pri#primorial_numbers">Index entries for sequences related to primorial numbers</a>
%e A346203 a(5) = 7 since 5 occurs in prime(7)# = 2 * 3 * 5 * 7 * 11 * 13 * 17 = 510510, but not in prime(0)#, prime(1)#, prime(2)#, ..., prime(6)#.
%t A346203 primorial[n_] := Product[Prime[j], {j, 1, n}]; a[n_] := (k = 0; While[! MatchQ[IntegerDigits[primorial[k]], {___, Sequence @@ IntegerDigits[n], ___}], k++]; k); Table[a[n], {n, 0, 70}]
%o A346203 (Python)
%o A346203 from sympy import nextprime
%o A346203 def A346203(n):
%o A346203     m, k, p, s = 1, 0, 1, str(n)
%o A346203     while s not in str(m):
%o A346203         k += 1
%o A346203         p = nextprime(p)
%o A346203         m *= p
%o A346203     return k # _Chai Wah Wu_, Jul 12 2021
%o A346203 (PARI) a(n) = my(k=0, p=1, q=1, sn=Str(n)); while (#strsplit(Str(q), sn)==1, k++; p=nextprime(p+1); q*=p); k; \\ _Michel Marcus_, Jul 13 2021; corrected Jun 15 2022
%Y A346203 Cf. A002110, A030000, A062584, A082058, A346120.
%K A346203 nonn,base
%O A346203 0,1
%A A346203 _Ilya Gutkovskiy_, Jul 10 2021