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A368681
Products of primorials that are perfect powers.
2
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
1,2
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
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[#]] &]
KEYWORD
nonn
AUTHOR
Michael De Vlieger, Jan 02 2024
STATUS
approved