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A328285 Smallest positive number k >= 2 for which there exist exactly n >= 1 integers m in M = {1, 2, 3, ..., k-1} such that k*m is a perfect power.

%I A328285 #25 Sep 08 2022 08:46:24
%S A328285 12,4,8,48,16,32,49,640,108,162,64,121,243,144,196,225,867,289,324,
%T A328285 361,256,400,484,529,512,1250,676,625,576,1682,784,900,961,1458,729,
%U A328285 1156,1225,2312,1369,1024,1521,2048,1681,1600,1849,1936,6348,2025,2209,4232
%N A328285 Smallest positive number k >= 2 for which there exist exactly n >= 1 integers m in M = {1, 2, 3, ..., k-1} such that k*m is a perfect power.
%H A328285 Chai Wah Wu, <a href="/A328285/b328285.txt">Table of n, a(n) for n = 1..763</a>
%e A328285 For n = 1 and k = 12 the set M = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11} and we obtain only 12 * 3 = 36 = 6^2, so a(1) = 12.
%e A328285 For n = 2 and k = 4 the set M = {1, 2, 3} and we obtain 4 * 1 = 4 = 2^2 and 4 * 2 = 8 = 2^3 so a(2) = 4.
%e A328285 For n = 3 and k = 8 the set M = {1, 2, 3, 4, 5, 6, 7}. The powers 8 * 1 = 2^3, 8 * 2 = 16 = 2^4 and 8 * 4 = 32 = 2^5 are obtained, so a(3) = 8.
%t A328285 ppQ[n_] := 1 < GCD @@ FactorInteger[n][[All, 2]]; cnt[k_] := cnt[k] = Length[ Select[ Range[k-1], ppQ[k #] &]]; a[n_] := Block[{k = n + 1}, While[ cnt[k] != n, k++]; k]; Array[a, 40] (* _Giovanni Resta_, Dec 05 2019 *)
%o A328285 (Magma) a:=[]; for n in [1..40] do k:=1; while #[m:m in [1..k-1]| IsPower(m*k)] ne n do k:=k+1; end while; Append(~a,k); end for; a;
%o A328285 (PARI) a(n) = {my(k=2); while (sum(m=1, k-1, ispower(m*k) != 0) != n, k++); k;} \\ _Michel Marcus_, Dec 05 2019
%Y A328285 Cf. A001597, A133466, A195085.
%K A328285 nonn
%O A328285 1,1
%A A328285 _Marius A. Burtea_, Nov 29 2019