cp's OEIS Frontend

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.

%I A355735 #11 Jul 24 2022 14:13:54
%S A355735 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,
%T A355735 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,
%U A355735 5,4,6,3,4,6,4,4,4,7,4,2,5,2,2,5,3,4,7
%N 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.
%C A355735 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 A355735 Wikipedia, <a href="https://en.wikipedia.org/wiki/Cartesian_product">Cartesian product</a>.
%e A355735 The a(15) = 3 ways are: (1,1), (1,3), (2,3).
%e A355735 The a(18) = 3 ways are: (1,1,1), (1,1,2), (1,2,2).
%e A355735 The a(2) = 1 through a(19) = 4 ways:
%e A355735   1  1  11  1  11  1  111  11  11  1  111  1  11  11  1111  1  111  1
%e A355735      2      3  12  2       12  13  5  112  2  12  13        7  112  2
%e A355735                    4       22              3  14  23           122  4
%e A355735                                            6                        8
%t A355735 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A355735 Table[Length[Select[Tuples[Divisors/@primeMS[n]],LessEqual@@#&]],{n,100}]
%Y A355735 Allowing any choice of divisors gives A355731, firsts A355732.
%Y A355735 Choosing a multiset instead of sequence gives A355733, firsts A355734.
%Y A355735 Positions of first appearances are A355736.
%Y A355735 Choosing only prime divisors gives A355745, variations A355741, A355744.
%Y A355735 The reverse version is A355749.
%Y A355735 A000005 counts divisors.
%Y A355735 A001414 adds up distinct prime divisors, counted by A001221.
%Y A355735 A003963 multiplies together the prime indices of n.
%Y A355735 A056239 adds up prime indices, row sums of A112798, counted by A001222.
%Y A355735 A061395 selects the maximum prime index.
%Y A355735 A120383 lists numbers divisible by all of their prime indices.
%Y A355735 A324850 lists numbers divisible by the product of their prime indices.
%Y A355735 Cf. A000720, A076610, A316524, A335433, A335448, A340827, A340852, A344616, A355737, A355739, A355740, A355742.
%K A355735 nonn
%O A355735 1,3
%A A355735 _Gus Wiseman_, Jul 16 2022