A368507 Powers of superprimorials.

1, 2, 4, 8, 12, 16, 32, 64, 128, 144, 256, 360, 512, 1024, 1728, 2048, 4096, 8192, 16384, 20736, 32768, 65536, 75600, 129600, 131072, 248832, 262144, 524288, 1048576, 2097152, 2985984, 4194304, 8388608, 16777216, 33554432, 35831808, 46656000, 67108864, 134217728

Offset: 1

Author

Michael De Vlieger, Dec 28 2023

Keywords

• nonn
• easy

Comments

Numbers k = Product_{i=1..j} p_i^e_(m*(j-i+1)) for m >= 0 and j >= 1.

Let b(n) = A006939(n) and let P(n) = A002110(n).

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).

The only prime in this sequence is 2.

Prime powers in this sequence are powers of 2.

Outside of {1, 2}, superprimorials are in A364710.

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.

Examples

			Powers of 2 are in the sequence since 2 = P(1).
Powers of 12 are terms, since 12 = P(1)*P(2).
Powers of 360 are terms, since 360 = P(1)*P(2)*P(3), etc.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..5301

Crossrefs

Cf. A000079, A002110, A006939, A025487, A126706, A246547, A286708, A364710, A364930, A368508.

Programs

Mathematica

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]]