A368682 Products of primorials that are perfect powers but not prime powers.

36, 144, 216, 576, 900, 1296, 1728, 2304, 3600, 5184, 7776, 9216, 13824, 14400, 20736, 27000, 32400, 36864, 44100, 46656, 57600, 82944, 110592, 129600, 147456, 176400, 186624, 216000, 230400, 248832, 279936, 331776, 373248, 518400, 589824, 705600, 746496, 810000

Offset: 1

Author

Michael De Vlieger, Jan 02 2024

Keywords

• nonn

Comments

Intersection of A025487 and A131605.

Proper subset of A286708.

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

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

Examples

			b(n) = A025487(n).
a(1) = b(11) = 36 = 6^2 = b(4)^2,
a(2) = b(19) = 144 = 12^2 = b(6)^2,
a(3) = b(23) = 216 = 6^3 = b(4)^3,
a(4) = b(33) = 576 = 24^2 = b(8)^2,
a(5) = b(38) = 900 = 30^2 = b(9)^2, etc.
		

Links

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

Crossrefs

Cf. A000079, A000961, A001597, A002110, A025487, A131605, A286708, A365308, A368508, A368681.

Programs

Mathematica

Select[Range[36, 2^18, 2], And[Union@ Differences@ PrimePi@ #1 == {1}, AllTrue[Union@ Differences@ #2, # <= 0 &], GCD @@ #2 > 1] & @@ Transpose@ FactorInteger[#] &]

Formula

This sequence is { A368681 \ A000079 }.