A062584 -id:A062584 - cp's OEIS frontend

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

Ilya Gutkovskiy, Jul 10 2021

			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)#.

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

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

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}]

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

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

A381099 a(n) is the smallest prime number that contains Fibonacci(n) as a substring.

Original entry on oeis.org

101, 11, 11, 2, 3, 5, 83, 13, 211, 347, 557, 89, 1447, 233, 2377, 6101, 1987, 1597, 25841, 24181, 67651, 109469, 177113, 28657, 2463683, 1750253, 1213931, 1964189, 2317811, 514229, 8320409, 13462693, 22178309, 35245781, 135702887, 192274651, 149303521

Offset: 0

Gonzalo Martínez, Feb 13 2025

If Fibonacci(n) is itself prime (n in A001605), then a(n) = Fibonacci(n).

			For n = 6, since Fibonacci(6) = 8, the smallest prime that contains 8 as a substring is 83, so a(6) = 83.

Cf. A000040, A000045, A001605, A062584.

a(n) = A062584(A000045(n)).

A382392 a(n) is the least prime number whose factorial base expansion contains the digit n.

Original entry on oeis.org

2, 2, 5, 19, 97, 601, 4327, 35281, 322571, 3265949, 36288017, 439084817, 5748019201, 80951270459, 1220496076831, 19615115520037, 334764638208037, 6046686277632071, 115242726703104073, 2311256907767808001, 48658040163532800037, 1072909785605898240031

Offset: 0

Rémy Sigrist, Mar 23 2025

This sequence is well defined: a(0) = a(1) = 2, and for n > 1, (n+1)! and n*n! + 1 are coprime, so by Dirichlet's theorem on arithmetic progressions, there exists a prime number p of the form k*(n+1)! + n*n! + 1 for some k >= 0, and the factorial base expansion of this prime number contains the digit n, hence a(n) <= p.

			The initial terms, in decimal and in factorial base, are:
  n  a(n)     fact(a(n))
  -  -------  -----------------
  0        2  1,0
  1        2  1,0
  2        5  2,1
  3       19  3,0,1
  4       97  4,0,0,1
  5      601  5,0,0,0,1
  6     4327  6,0,0,1,0,1
  7    35281  7,0,0,0,0,0,1
  8   322571  8,0,0,0,0,1,2,1
  9  3265949  9,0,0,0,0,1,0,2,1

Index entries for sequences related to factorial base representation

Cf. A001563, A062584, A090703.

a(n) = { forprime (p = n*n!, oo, my (q = p); for (r = 2, oo, if (q==0, break, q % r==n, return (p), q \= r););); }

a(n) > A001563(n).

cp's OEIS Frontend

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.

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A381099 a(n) is the smallest prime number that contains Fibonacci(n) as a substring.

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A382392 a(n) is the least prime number whose factorial base expansion contains the digit n.

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