A036507 Smallest square containing exactly n decimal digits '0'.
0, 100, 102400, 10000, 10240000, 1000000, 1024000000, 100000000, 102400000000, 10000000000, 10240000000000, 1000000000000, 1024000000000000, 100000000000000, 102400000000000000, 10000000000000000, 10240000000000000000, 1000000000000000000, 1024000000000000000000
Offset: 1
Crossrefs
Programs
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Mathematica
nsmall = Table[Infinity, 20]; For[i = 0, i <= 4*10^6, i++, n0 = Count[IntegerDigits[i^2], 0]; If[nsmall[[n0]] > i^2, nsmall[[n0]] = i^2]]; ReplaceAll[nsmall, Infinity -> "?"] (* Robert Price, Mar 22 2020 *) a[n_] := If[OddQ[n], 1024*10^(n-1), 10^n]; a[1] = 0; Array[a, 20] (* Amiram Eldar, Aug 26 2025 *)
Formula
a(2*n) = 10^(2*n), a(2*n+1) = 1024*10^(2*n) for n >= 1 since 1024 is the smallest square factor that contains a single '0'. - Georg Fischer, Jul 03 2023
Sum_{n>=2} 1/a(n) = 1025/101376. - Amiram Eldar, Aug 26 2025