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a(n) is the smallest number whose factorial has 10^n trailing zeros.
3

%I #11 Aug 22 2017 15:08:43

%S 5,45,405,4005,40010,400005,4000005,40000010,400000015,4000000015,

%T 40000000015,400000000015,4000000000010,40000000000015,

%U 400000000000020,4000000000000025,40000000000000025,400000000000000025

%N a(n) is the smallest number whose factorial has 10^n trailing zeros.

%H A. M. Oller-Marcen, J. Maria Grau, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Oller/oller3.html">On the Base-b Expansion of the Number of Trailing Zeros of b^k!</a>, J. Int. Seq. 14 (2011) 11.6.8

%e 4!=24 has no zeros, and 5!=120 has 1 (10^0) zero ==> a(0)= 5.

%e 44!=2658271574788448768043625811014615890319638528000000000 (9 zeros) and 45!=119622220865480194561963161495657715064383733760000000000 (10^1= 10 zeros) ==> a(1)=45.

%t Z10[n_] := Floor[Sum[Floor[n/5^i], {i, 1, Floor[Log[5, n]]}]]; a[n_] := Module[{m = 4*10^n}, While[Z10[m] < 10^n, m++ ]; m]; Table[a[n], {n, 0, 60}]

%Y If A173292(n)> -1 then a(n)=A173292(n)

%K base,nonn

%O 0,1

%A _José María Grau Ribas_, Feb 21 2010