A379944 Smallest number of leading digits of n! that form a prime (or 0 if none exist).

0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 2, 0, 0, 2, 1, 1, 0, 7, 1, 1, 2, 1, 0, 5, 0, 0, 1, 8, 1, 0, 1, 6, 1, 3, 1, 2, 1, 1, 0, 1, 8, 38, 1, 2, 1, 1, 5, 34, 1, 5, 6, 0, 1, 0, 1, 6, 1, 2, 2, 1, 1, 2, 8, 9, 1, 1, 1, 2, 2, 0, 2, 5, 1, 1, 0, 4, 2, 2, 1, 1, 2, 1, 1, 1, 1

Offset

0,13

Comments

It appears that as n gets large, a(n) can become arbitrarily large.

It appears that values of n such that a(n) = 0 exist for arbitrarily large n.

Links

Carson R. Smith, Table of n, a(n) for n = 0..2000

Example

For n = 3, 3! = 6, 6 is not prime, a(3) = 0.
For n = 19, 19! = 121645100408832000, 1216451 is the smallest prime, a(19) = 7.

Mathematica

A379944[n_] := Catch[Do[If[PrimeQ[FromDigits[#[[;; k]]]], Throw[k]], {k, Length[#]}] & [IntegerDigits[n!]]; 0];
Array[A379944, 100, 0] (* Paolo Xausa, Jan 16 2025 *)

Prog

(Python)
import math
from sympy import isprime
def a(n):
    factorial = str(math.factorial(n))
    for d in range(1, len(factorial)+1):
        if isprime(int(factorial[:d])):
            return d
    return 0
(PARI) a(n) = my(d=digits(n!)); for (k=1, #d, if (isprime(fromdigits(Vec(d, k))), return(k))); \\ Michel Marcus, Jan 08 2025

Crossrefs

Cf. A000040, A000142, A046277.

Sequence in context: A029304 A030202 A159818 * A081827 A100286 A280912

Adjacent sequences: A379941 A379942 A379943 * A379945 A379946 A379947

Keyword

nonn,base,new

Author

Carson R. Smith, Jan 07 2025

Extensions

More terms from Jinyuan Wang, Jan 07 2025

Status

approved