cp's OEIS Frontend

A355734 Least k such that there are exactly n multisets that can be obtained by choosing a divisor of each prime index of k.

%I A355734 #7 Jul 22 2022 20:52:04
%S A355734 1,3,7,13,21,35,39,89,133,105,91,195,351,285,247,333,273,481,455,555,
%T A355734 623,801,791,741,1359,1157,1281,1335,1365,1443,1977,1729,1967,1869,
%U A355734 2109,3185,2373,2769,2639,4361,3367,3653,3885,3471,4613,5883,5187,5551,6327
%N A355734 Least k such that there are exactly n multisets that can be obtained by choosing a divisor of each prime index of k.
%C A355734 This is the position of first appearance of n in A355733.
%C A355734 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 A355734 The terms together with their prime indices begin:
%e A355734     1: {}
%e A355734     3: {2}
%e A355734     7: {4}
%e A355734    13: {6}
%e A355734    21: {2,4}
%e A355734    35: {3,4}
%e A355734    39: {2,6}
%e A355734    89: {24}
%e A355734   133: {4,8}
%e A355734   105: {2,3,4}
%e A355734    91: {4,6}
%e A355734   195: {2,3,6}
%e A355734   351: {2,2,2,6}
%e A355734 For example, the choices for a(12) = 195 are:
%e A355734   {1,1,1}  {1,2,2}  {1,3,6}
%e A355734   {1,1,2}  {1,2,3}  {2,2,3}
%e A355734   {1,1,3}  {1,2,6}  {2,3,3}
%e A355734   {1,1,6}  {1,3,3}  {2,3,6}
%t A355734 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A355734 mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
%t A355734 az=Table[Length[Union[Sort/@Tuples[Divisors/@primeMS[n]]]],{n,1000}];
%t A355734 Table[Position[az,k][[1,1]],{k,mnrm[az]}]
%Y A355734 Counting all choices of divisors gives A355732, firsts of A355731.
%Y A355734 Positions of first appearances in A355733.
%Y A355734 Choosing weakly increasing divisors gives A355736, firsts of A355735.
%Y A355734 A000005 counts divisors.
%Y A355734 A001414 adds up distinct prime divisors, counted by A001221.
%Y A355734 A003963 multiplies together the prime indices of n.
%Y A355734 A056239 adds up prime indices, row sums of A112798, counted by A001222.
%Y A355734 A120383 lists numbers divisible by all of their prime indices.
%Y A355734 A324850 lists numbers divisible by the product of their prime indices.
%Y A355734 Cf. A000720, A076610, A340852, A344606, A355737, A355739, A355740, A355741, A355744, A355747.
%K A355734 nonn
%O A355734 1,2
%A A355734 _Gus Wiseman_, Jul 21 2022