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

1,1

Links

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

Example

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.

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 *)

Prog

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

Crossrefs

Cf. A001597, A133466, A195085.

Sequence in context: A306375 A139358 A004477 * A307164 A181829 A339467

Adjacent sequences: A328282 A328283 A328284 * A328286 A328287 A328288

Keyword

nonn

Author

Marius A. Burtea, Nov 29 2019

Status

approved