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

1,2

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.

Links

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

Example

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.

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]]

Crossrefs

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

Sequence in context: A375599 A324214 A371732 * A273109 A046843 A376256

Adjacent sequences: A368504 A368505 A368506 * A368508 A368509 A368510

Keyword

nonn,easy

Author

Michael De Vlieger, Dec 28 2023

Status

approved