A360903 a(n) is the least number that has exactly 2^n squarefree divisors and exactly 2^n powerful divisors.

1, 4, 36, 720, 25200, 1940400, 227026800, 42454011600, 10486140865200, 3858899838393600, 1902437620328044800, 1120535758373218387200, 953575930375608847507200, 977415328634999068694880000, 1218836914807843838662515360000, 1775845384875028472931284879520000

Offset: 0

Author

Amiram Eldar, Feb 25 2023

Keywords

• nonn

Comments

a(n) is the least term k of A360902 with A034444(k) = A005361(k) = 2^n.

Examples

			a(1) = 4 since 4 is the least number that has 2^1 = 2 squarefree divisors (1 and 2) and 2 powerful divisors (1 and 4).
a(2) = 36 since 36 is the least number that has 2^2 = 4 squarefree divisors (1, 2, 3 and 6) and 4 powerful divisors (1, 4, 9 and 36).
		

Links

• Amiram Eldar, Table of n, a(n) for n = 0..18

Crossrefs

Subsequence of A025487 and A360902.

Cf. A005361, A034444.

Programs

Mathematica

f1[p_, e_] := 2; s1[1] = 1; s1[n_] := Times @@ f1 @@@ FactorInteger[n]; f2[p_, e_] := e; s2[1] = 1; s2[n_] := Times @@ f2 @@@ FactorInteger[n]; v = Cases[Import["https://oeis.org/A025487/b025487.txt", "Table"], {, }][[;; , 2]]; With[{m = 9}, seq = Table[0, {m}]; Do[If[(s = s1[v[[k]]]) == s2[v[[k]]], e = IntegerExponent[s, 2] + 1; If[e <= m && seq[[e]] == 0, seq[[e]] = v[[k]]]], {k, 1, Length[v]}]; seq]