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%I #5 Jun 18 2021 01:16:52
%S 13,7,25,59,193,313,1447,103,2431,1305,29089,21117,10525,26663,63271,
%T 25127,106501,215735,1143579,5115041,810163,9554677,23932393,13101403,
%U 14299679,62266699,739155479,37795511,4015916137,3197105709,5386711727,14904706741,20696039773
%N a(n) is the least number k such that {k, k^2, ..., k^n} are all odious numbers (A000069), but k^(n+1) is not.
%e 13 is a term since it is an odious number (13 = 1101_2 has 3 1's), but 13^2 = 169 is not (169 = 10101001_2 has 4 1's), and 13 is the least number with this property.
%t odQ[n_] := n > 1 && OddQ[n]; odiousQ[n_] := odQ @ DigitCount[n, 2, 1]; f[n_] := Module[{e = 0, r = n}, While[odiousQ[r], r *= n; e++]; e]; m = 15; s = Table[0, {m}]; c = 0; n = 1; While[c < m, k = f[n]; If[0 < k <= m && s[[k]] == 0, c++; s[[k]] = n]; n++]; s
%Y Cf. A000069, A345398.
%K nonn,base
%O 1,1
%A _Amiram Eldar_, Jun 17 2021