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

0,2

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).

Links

Table of n, a(n) for n=0..16.

Example

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.

Prog

(Python 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
(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

Crossrefs

Cf. A027868, A000966.

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

Cf. A252652.

Sequence in context: A070004 A104356 A137935 * A115182 A313762 A313763

Adjacent sequences: A255397 A255398 A255399 * A255401 A255402 A255403

Keyword

nonn,base,more

Author

Martin Y. Champel, Feb 22 2015

Status

approved