A116898 Numbers k such that k! is turned into a prime number by changing its trailing 0's into 1's.

2, 10, 34, 499, 1746

Offset: 1

Author

Giovanni Resta, Mar 07 2006

Keywords

• nonn
• base
• hard
• more

Comments

Also numbers k such that n! + R(Z(k)) is prime, where R(t) = (10^t - 1)/9 is the repunit with t digits (A002275) and Z(m) = Sum_{j>=1} floor(m/5^j) is the number of trailing zeros of m! (A027868). The (probable) prime corresponding to a(5)=1746 has 4905 digits. Next term must be greater than 4000.

Next term must be greater than 20000. - Michael S. Branicky, Nov 14 2024

Examples

			10 is a term, since 10! = 3628800 and 3628811 is prime.
		

Crossrefs

Cf. A000142, A002275, A027868.

Programs

Maple

q:= n-> (f-> isprime(f+(10^padic[ordp](f, 10)-1)/9))(n!):
select(q, [$1..500])[];  # Alois P. Heinz, Feb 10 2021

Mathematica

tz1Q[n_]:=Module[{idn=Split[IntegerDigits[n!]]},PrimeQ[ FromDigits[ Flatten[ Join[ Most[ idn], Last[idn]/.(0->1)]]]]]; Select[ Range[ 1800],tz1Q] (* Harvey P. Dale, Oct 01 2015 *)

Python

from sympy import isprime
from math import factorial
def ok(n):
  s, zeros = str(factorial(n)), 0
  while s[-1] == '0': s = s[:-1]; zeros += 1
  return isprime(int(s + '1'*zeros))
print([m for m in range(500) if ok(m)]) # Michael S. Branicky, Feb 10 2021