A355735 Number of ways to choose a divisor of each prime index of n (taken in weakly increasing order) such that the result is weakly increasing.

1, 1, 2, 1, 2, 2, 3, 1, 3, 2, 2, 2, 4, 3, 3, 1, 2, 3, 4, 2, 5, 2, 3, 2, 3, 4, 4, 3, 4, 3, 2, 1, 3, 2, 4, 3, 6, 4, 7, 2, 2, 5, 4, 2, 4, 3, 4, 2, 6, 3, 3, 4, 5, 4, 3, 3, 7, 4, 2, 3, 6, 2, 7, 1, 6, 3, 2, 2, 5, 4, 6, 3, 4, 6, 4, 4, 4, 7, 4, 2, 5, 2, 2, 5, 3, 4, 7

Offset: 1

Author

Gus Wiseman, Jul 16 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(15) = 3 ways are: (1,1), (1,3), (2,3).
The a(18) = 3 ways are: (1,1,1), (1,1,2), (1,2,2).
The a(2) = 1 through a(19) = 4 ways:
  1  1  11  1  11  1  111  11  11  1  111  1  11  11  1111  1  111  1
     2      3  12  2       12  13  5  112  2  12  13        7  112  2
                   4       22              3  14  23           122  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.

Positions of first appearances are A355736.

Choosing only prime divisors gives A355745, variations A355741, A355744.

The reverse version is A355749.

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.

A120383 lists numbers divisible by all of their prime indices.

A324850 lists numbers divisible by the product of their prime indices.

Cf. A000720, A076610, A316524, A335433, A335448, A340827, A340852, A344616, A355737, A355739, A355740, A355742.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[Tuples[Divisors/@primeMS[n]],LessEqual@@#&]],{n,100}]