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

1,6

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.

Links

Table of n, a(n) for n=1..46.

Example

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)

Mathematica

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

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.

Sequence in context: A010334 A010578 A328583 * A240674 A005866 A125584

Adjacent sequences: A355534 A355535 A355536 * A355538 A355539 A355540

Keyword

nonn

Author

Gus Wiseman, Jul 20 2022

Status

approved