cp's OEIS Frontend

A355736 Least k such that there are exactly n ways to choose a divisor of each prime index of k (taken in weakly increasing order) such that the result is also weakly increasing.

%I A355736 #7 Jul 22 2022 20:51:59
%S A355736 1,3,7,13,21,37,39,89,133,117,111,273,351,259,267,333,453,793,669,623,
%T A355736 999,777,843,1491,1157,1561,2863,1443,1963,2331,1977,1869,2899,2529,
%U A355736 3207,4107,3171,5073,4329,3653,4667,3471,7399,4613,7587,5931,7269,5889,7483
%N A355736 Least k such that there are exactly n ways to choose a divisor of each prime index of k (taken in weakly increasing order) such that the result is also weakly increasing.
%C A355736 This is the position of first appearance of n in A355735.
%C A355736 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.
%e A355736 The terms together with their prime indices begin:
%e A355736      1: {}
%e A355736      3: {2}
%e A355736      7: {4}
%e A355736     13: {6}
%e A355736     21: {2,4}
%e A355736     37: {12}
%e A355736     39: {2,6}
%e A355736     89: {24}
%e A355736    133: {4,8}
%e A355736    117: {2,2,6}
%e A355736    111: {2,12}
%e A355736    273: {2,4,6}
%e A355736    351: {2,2,2,6}
%e A355736 For example, the choices for a(12) = 273 are:
%e A355736   {1,1,1}  {1,2,2}  {2,2,2}
%e A355736   {1,1,2}  {1,2,3}  {2,2,3}
%e A355736   {1,1,3}  {1,2,6}  {2,2,6}
%e A355736   {1,1,6}  {1,4,6}  {2,4,6}
%t A355736 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A355736 mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
%t A355736 az=Table[Length[Select[Tuples[Divisors/@primeMS[n]],LessEqual@@#&]],{n,1000}];
%t A355736 Table[Position[az,k][[1,1]],{k,mnrm[az]}]
%Y A355736 Allowing any choice of divisors gives A355732, firsts of A355731.
%Y A355736 Choosing a multiset instead of sequence gives A355734, firsts of A355733.
%Y A355736 Positions of first appearances in A355735.
%Y A355736 The case of prime factors instead of divisors is counted by A355745.
%Y A355736 The decreasing version is counted by A355749.
%Y A355736 A000005 counts divisors.
%Y A355736 A001414 adds up distinct prime divisors, counted by A001221.
%Y A355736 A003963 multiplies together the prime indices of n.
%Y A355736 A056239 adds up prime indices, row sums of A112798, counted by A001222.
%Y A355736 A120383 lists numbers divisible by all of their prime indices.
%Y A355736 A324850 lists numbers divisible by the product of their prime indices.
%Y A355736 Cf. A000720, A076610, A355737, A355739, A355740, A355741, A355744.
%K A355736 nonn
%O A355736 1,2
%A A355736 _Gus Wiseman_, Jul 21 2022