A355749 - OEIS

%I #5 Jul 19 2022 08:03:43

%S 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,

%T 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,

%U 2,1,6,1,4,1,2,1,3,1,4,1,5,1,2,1,2,1,3

%N 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).

%C 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.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Cartesian_product">Cartesian product</a>.

%e The a(2) = 1 through a(19) = 4 choices:

%e   1  1  11  1  11  1  111  11  11  1  111  1  11  11  1111  1  111  1

%e      2      3      2       21      5       2      21        7       2

%e                    4       22              3                        4

%e                                            6                        8

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

%Y Allowing any choice of divisors gives A355731, firsts A355732.

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

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

%Y A000005 counts divisors.

%Y A001414 adds up distinct prime divisors, counted by A001221.

%Y A003963 multiplies together the prime indices of n.

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

%Y A061395 selects the maximum prime index.

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

%K nonn

%O 1,3

%A _Gus Wiseman_, Jul 18 2022