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.

12, 4, 8, 48, 16, 32, 49, 640, 108, 162, 64, 121, 243, 144, 196, 225, 867, 289, 324, 361, 256, 400, 484, 529, 512, 1250, 676, 625, 576, 1682, 784, 900, 961, 1458, 729, 1156, 1225, 2312, 1369, 1024, 1521, 2048, 1681, 1600, 1849, 1936, 6348, 2025, 2209, 4232

Offset: 1

Author

Marius A. Burtea, Nov 29 2019

Keywords

• nonn

Examples

			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.
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.
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.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..763

Crossrefs

Cf. A001597, A133466, A195085.

Programs

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;

Mathematica

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 *)

PARI

a(n) = {my(k=2); while (sum(m=1, k-1, ispower(m*k) != 0) != n, k++); k;} \\ Michel Marcus, Dec 05 2019