A368681 Products of primorials that are perfect powers.

1, 4, 8, 16, 32, 36, 64, 128, 144, 216, 256, 512, 576, 900, 1024, 1296, 1728, 2048, 2304, 3600, 4096, 5184, 7776, 8192, 9216, 13824, 14400, 16384, 20736, 27000, 32400, 32768, 36864, 44100, 46656, 57600, 65536, 82944, 110592, 129600, 131072, 147456, 176400, 186624

Offset: 1

Author

Michael De Vlieger, Jan 02 2024

Keywords

• nonn

Comments

Intersection of A025487 and A001597.

Contains A365308 (perfect powers of composite primorials), A368508 (perfect powers of composite superprimorials), and A368682.

These numbers are perfect powers of some smaller product of primorials.

Examples

			Let b(n) = A025487(n).
a(1) = b(1) = 1 = 1^k = b(1)^k, k >= 2,
a(2) = b(3) = 4 = 2^2 = b(2)^2,
a(3) = b(5) = 8 = 2^3 = b(2)^3,
a(6) = b(11) = 36 = 6^2 = b(4)^2,
a(9) = b(19) = 144 = 12^2 = b(6)^2, etc.
2 is not in the sequence since 2 is squarefree and not in A001597.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000079, A001597, A002110, A025487, A131605, A365308, A368508, A368682.

Programs

Mathematica

{1}~Join~Select[Range[4, 200000, 2], Or[PrimePowerQ[#], And[Union@ Differences@ PrimePi@ #1 == {1}, AllTrue[Union@ Differences@ #2, # <= 0 &], GCD @@ #2 > 1] & @@ Transpose@ FactorInteger[#]] &]