A355537 Number of ways to choose a sequence of prime factors, one of each integer from 2 to n.

1, 1, 1, 1, 1, 2, 2, 2, 2, 4, 4, 8, 8, 16, 32, 32, 32, 64, 64, 128, 256, 512, 512, 1024, 1024, 2048, 2048, 4096, 4096, 12288, 12288, 12288, 24576, 49152, 98304, 196608, 196608, 393216, 786432, 1572864, 1572864, 4718592, 4718592, 9437184, 18874368, 37748736

Offset: 1

Author

Gus Wiseman, Jul 20 2022

Keywords

• nonn

Comments

Also partial products of A001221 without the first term 0, sum A013939.

For initial terms up to n = 29 we have a(n) = 2^A355538(n). The first non-power of 2 is a(30) = 12288.

Examples

			The a(n) choices for n = 2, 6, 10, 12, with prime(k) replaced by k:
  (1)  (12131)  (121314121)  (12131412151)
       (12132)  (121314123)  (12131412152)
                (121324121)  (12131412351)
                (121324123)  (12131412352)
                             (12132412151)
                             (12132412152)
                             (12132412351)
                             (12132412352)
		

Crossrefs

The sum of the same integers is A000096.

The product of the same integers is A000142, Heinz number A070826.

The version for divisors instead of prime factors is A066843.

The integers themselves are the rows of A131818.

The version with multiplicity is A327486.

Using prime indices instead of 2..n gives A355741, for multisets A355744.

Counting sequences instead of multisets gives A355746.

A001221 counts distinct prime factors, with sum A001414.

A001222 counts prime factors with multiplicity.

A003963 multiplies together the prime indices of n.

A056239 adds up prime indices, row sums of A112798.

Cf. A000005, A000040, A000720, A002110, A013939, A076610, A355538, A355731, A355733, A355745, A355747.

Programs

Mathematica

Table[Times@@PrimeNu/@Range[2,m],{m,2,30}]