A368681 - OEIS

%I #7 Jan 03 2024 03:37:03

%S 1,4,8,16,32,36,64,128,144,216,256,512,576,900,1024,1296,1728,2048,

%T 2304,3600,4096,5184,7776,8192,9216,13824,14400,16384,20736,27000,

%U 32400,32768,36864,44100,46656,57600,65536,82944,110592,129600,131072,147456,176400,186624

%N Products of primorials that are perfect powers.

%C Intersection of A025487 and A001597.

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

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

%H Michael De Vlieger, <a href="/A368681/b368681.txt">Table of n, a(n) for n = 1..10000</a>

%e Let b(n) = A025487(n).

%e a(1) = b(1) = 1 = 1^k = b(1)^k, k >= 2,

%e a(2) = b(3) = 4 = 2^2 = b(2)^2,

%e a(3) = b(5) = 8 = 2^3 = b(2)^3,

%e a(6) = b(11) = 36 = 6^2 = b(4)^2,

%e a(9) = b(19) = 144 = 12^2 = b(6)^2, etc.

%e 2 is not in the sequence since 2 is squarefree and not in A001597.

%t {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[#]] &]

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

%K nonn

%O 1,2

%A _Michael De Vlieger_, Jan 02 2024