A368681 - OEIS

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

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OFFSET

1,2

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.

LINKS

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

EXAMPLE

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.

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[#]] &]

CROSSREFS

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

Sequence in context: A075090 A354955 A304250 * A322793 A283052 A088259

Adjacent sequences: A368678 A368679 A368680 * A368682 A368683 A368684

KEYWORD

nonn

AUTHOR

Michael De Vlieger, Jan 02 2024

STATUS

approved