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.

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, 13, 9, 21, 3, 5, 10, 14, 12, 13, 26, 24, 12, 17, 18, 15, 12, 26, 16, 22, 10, 16, 12, 11, 13, 7, 13, 20, 17, 19, 11, 20, 15, 18, 11, 14, 21, 13, 10, 24, 20, 14, 21, 8, 9

Offset: 0

Author

Ilya Gutkovskiy, Jul 10 2021

Keywords

• nonn
• base

Examples

			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)#.
		

Links

• Ilya Gutkovskiy, Scatterplot of a(n) up to n=100000
• Index entries for sequences related to primorial numbers

Crossrefs

Cf. A002110, A030000, A062584, A082058, A346120.

Programs

Mathematica

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}]

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

Python

from sympy import nextprime
def A346203(n):
    m, k, p, s = 1, 0, 1, str(n)
    while s not in str(m):
        k += 1
        p = nextprime(p)
        m *= p
    return k # Chai Wah Wu, Jul 12 2021