A355749 Number of ways to choose a weakly decreasing sequence of divisors, one of each prime index of n (with multiplicity, taken in weakly increasing order).

1, 1, 2, 1, 2, 1, 3, 1, 3, 1, 2, 1, 4, 1, 2, 1, 2, 1, 4, 1, 3, 1, 3, 1, 3, 1, 4, 1, 4, 1, 2, 1, 2, 1, 3, 1, 6, 1, 3, 1, 2, 1, 4, 1, 3, 1, 4, 1, 6, 1, 2, 1, 5, 1, 2, 1, 3, 1, 2, 1, 6, 1, 4, 1, 4, 1, 2, 1, 2, 1, 6, 1, 4, 1, 2, 1, 3, 1, 4, 1, 5, 1, 2, 1, 2, 1, 3

Offset: 1

Author

Gus Wiseman, Jul 18 2022

Keywords

• nonn

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

Examples

			The a(2) = 1 through a(19) = 4 choices:
  1  1  11  1  11  1  111  11  11  1  111  1  11  11  1111  1  111  1
     2      3      2       21      5       2      21        7       2
                   4       22              3                        4
                                           6                        8
		

Links

• Wikipedia, Cartesian product.

Crossrefs

Allowing any choice of divisors gives A355731, firsts A355732.

Choosing a multiset instead of sequence gives A355733, firsts A355734.

The reverse version is A355735, firsts A355736, only primes A355745.

A000005 counts divisors.

A001414 adds up distinct prime divisors, counted by A001221.

A003963 multiplies together the prime indices of n.

A056239 adds up prime indices, row sums of A112798, counted by A001222.

A061395 selects the maximum prime index.

Cf. A000720, A076610, A120383, A316524, A324850, A355737, A355739, A355740, A355741, A355742, A355744.

Programs

Mathematica

Table[Length[Select[Tuples[Divisors/@primeMS[n]], GreaterEqual@@#&]],{n,100}]