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A368507 Powers of superprimorials.

%I A368507 #8 Dec 30 2023 23:49:28
%S A368507 1,2,4,8,12,16,32,64,128,144,256,360,512,1024,1728,2048,4096,8192,
%T A368507 16384,20736,32768,65536,75600,129600,131072,248832,262144,524288,
%U A368507 1048576,2097152,2985984,4194304,8388608,16777216,33554432,35831808,46656000,67108864,134217728
%N A368507 Powers of superprimorials.
%C A368507 Numbers k = Product_{i=1..j} p_i^e_(m*(j-i+1)) for m >= 0 and j >= 1.
%C A368507 Let b(n) = A006939(n) and let P(n) = A002110(n).
%C A368507 This sequence contains {1}, A000079, A006939, certain k in A364710 (intersection of A126706 and A025487), and certain m in A364930 (intersection of A286708 and A025487).
%C A368507 The only prime in this sequence is 2.
%C A368507 Prime powers in this sequence are powers of 2.
%C A368507 Outside of {1, 2}, superprimorials are in A364710.
%C A368507 Squareful numbers in this sequence contain {2^k, k > 1}, which are in A000079, a proper subset of A246547, and {b(k)^m, k > 1, m > 1}, which are in A364930, a proper subset of A286708.
%H A368507 Michael De Vlieger, <a href="/A368507/b368507.txt">Table of n, a(n) for n = 1..5301</a>
%e A368507 Powers of 2 are in the sequence since 2 = P(1).
%e A368507 Powers of 12 are terms, since 12 = P(1)*P(2).
%e A368507 Powers of 360 are terms, since 360 = P(1)*P(2)*P(3), etc.
%t A368507 nn = 2^60; k = 1; P = 2; Q = 2; {1}~Join~Union@ Reap[While[j = 1; While[Q^j < nn, Sow[Q^j]; j++]; j > 1, k++; P *= Prime[k]; Q *= P] ][[-1, 1]]
%Y A368507 Cf. A000079, A002110, A006939, A025487, A126706, A246547, A286708, A364710, A364930, A368508.
%K A368507 nonn,easy
%O A368507 1,2
%A A368507 _Michael De Vlieger_, Dec 28 2023