cp's OEIS Frontend

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

%I A255400 #49 Jan 15 2025 20:09:47
%S A255400 0,5,10,15,20,264,25,30,35,40,45,101805,50,55,60,65,70
%N A255400 a(n) is the smallest nonnegative integer such that a(n)! contains a string of exactly n consecutive 0's.
%C A255400 Most multiples of 5 belong to the sequence (if not all).
%C A255400 All terms whose indices are included in A000966 are far bigger than their neighboring terms whose indices are multiples of 5.
%C A255400 a(11) is a multiple of 5, we can verify a(11) = a(25448).
%e A255400 a(0) = 0 as 0! = 1 does not contain '0'.
%e A255400 a(1) = 5 as 5! = 120 contains '0'.
%e A255400 a(2) = 10 as 10! = 3628800 contains '00' and 10 is the smallest integer for which the condition is met.
%o A255400 (Python) # Python version 2.7
%o A255400 from math import factorial as fct
%o A255400 def trailing_zero(n):
%o A255400     k=0
%o A255400     while n!=0:
%o A255400         n/=5
%o A255400         k+=n
%o A255400     return k
%o A255400 def A255400():
%o A255400     index = 1
%o A255400     f = 1
%o A255400     while True:
%o A255400         if trailing_zero(f) == index:
%o A255400             print("A255400("+str(index)+") = " +str(f))
%o A255400             index += 1
%o A255400         elif trailing_zero(f) > index:
%o A255400             while True:
%o A255400                 clnzer = str(fct(f))[:-trailing_zero(f)]
%o A255400                 if index*'0' in clnzer and (index+1)*'0' not in clnzer:
%o A255400                     print("A255400("+str(index)+") = " +str(f))
%o A255400                     index += 1
%o A255400                     f = 0
%o A255400                     break
%o A255400                 f +=1
%o A255400         f +=1
%o A255400     return
%o A255400 (Python)
%o A255400 import re
%o A255400 def A255400(n):
%o A255400     f, i, s = 1, 0, re.compile('[0-9]*[1-9]0{'+str(n)+'}[1-9][0-9]*')
%o A255400     while s.match(str(f)+'1') is None:
%o A255400         i += 1
%o A255400         f *= i
%o A255400     return i # _Chai Wah Wu_, Apr 02 2015
%o A255400 (PARI) \\ uses is() from A000966
%o A255400 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);
%o A255400 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
%Y A255400 Cf. A027868, A000966.
%Y A255400 Cf. A254042, A254447, A254448, A254449, A254500, A254501, A254502, A254716, A254717.
%Y A255400 Cf. A252652.
%K A255400 nonn,base,more
%O A255400 0,2
%A A255400 _Martin Y. Champel_, Feb 22 2015