A255400 a(n) is the smallest nonnegative integer such that a(n)! contains a string of exactly n consecutive 0's.

0, 5, 10, 15, 20, 264, 25, 30, 35, 40, 45, 101805, 50, 55, 60, 65, 70

Offset: 0

Author

Martin Y. Champel, Feb 22 2015

Keywords

• nonn
• base
• more

Comments

Most multiples of 5 belong to the sequence (if not all).

All terms whose indices are included in A000966 are far bigger than their neighboring terms whose indices are multiples of 5.

a(11) is a multiple of 5, we can verify a(11) = a(25448).

Examples

			a(0) = 0 as 0! = 1 does not contain '0'.
a(1) = 5 as 5! = 120 contains '0'.
a(2) = 10 as 10! = 3628800 contains '00' and 10 is the smallest integer for which the condition is met.
		

Crossrefs

Cf. A027868, A000966.

Cf. A254042, A254447, A254448, A254449, A254500, A254501, A254502, A254716, A254717.

Cf. A252652.

Programs

PARI

\\ uses is() from A000966
f(k, special, sz, sz1) = my(f=k!); if (special, s=Str(f/10^valuation(f, 10)), s=Str(k!)); #strsplit(s, sz) - #strsplit(s, sz1);
a(n) = if (n==0, return(0)); my(sz= concat(vector(n, k, "0")), sz1=concat(sz, "0"), k=1,special=is(n)); while (f(k, special, sz, sz1) != 1, k++); k; \\ Michel Marcus, Oct 25 2023

Python

# Python version 2.7
from math import factorial as fct
def trailing_zero(n):
    k=0
    while n!=0:
        n/=5
        k+=n
    return k
def A255400():
    index = 1
    f = 1
    while True:
        if trailing_zero(f) == index:
            print("A255400("+str(index)+") = " +str(f))
            index += 1
        elif trailing_zero(f) > index:
            while True:
                clnzer = str(fct(f))[:-trailing_zero(f)]
                if index*'0' in clnzer and (index+1)*'0' not in clnzer:
                    print("A255400("+str(index)+") = " +str(f))
                    index += 1
                    f = 0
                    break
                f +=1
        f +=1
    return

Python

import re
def A255400(n):
    f, i, s = 1, 0, re.compile('[0-9]*[1-9]0{'+str(n)+'}[1-9][0-9]*')
    while s.match(str(f)+'1') is None:
        i += 1
        f *= i
    return i # Chai Wah Wu, Apr 02 2015