A355734 - OEIS

%I #7 Jul 22 2022 20:52:04

%S 1,3,7,13,21,35,39,89,133,105,91,195,351,285,247,333,273,481,455,555,

%T 623,801,791,741,1359,1157,1281,1335,1365,1443,1977,1729,1967,1869,

%U 2109,3185,2373,2769,2639,4361,3367,3653,3885,3471,4613,5883,5187,5551,6327

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

%C This is the position of first appearance of n in A355733.

%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.

%e The terms together with their prime indices begin:

%e     1: {}

%e     3: {2}

%e     7: {4}

%e    13: {6}

%e    21: {2,4}

%e    35: {3,4}

%e    39: {2,6}

%e    89: {24}

%e   133: {4,8}

%e   105: {2,3,4}

%e    91: {4,6}

%e   195: {2,3,6}

%e   351: {2,2,2,6}

%e For example, the choices for a(12) = 195 are:

%e   {1,1,1}  {1,2,2}  {1,3,6}

%e   {1,1,2}  {1,2,3}  {2,2,3}

%e   {1,1,3}  {1,2,6}  {2,3,3}

%e   {1,1,6}  {1,3,3}  {2,3,6}

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];

%t az=Table[Length[Union[Sort/@Tuples[Divisors/@primeMS[n]]]],{n,1000}];

%t Table[Position[az,k][[1,1]],{k,mnrm[az]}]

%Y Counting all choices of divisors gives A355732, firsts of A355731.

%Y Positions of first appearances in A355733.

%Y Choosing weakly increasing divisors gives A355736, firsts of A355735.

%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 A120383 lists numbers divisible by all of their prime indices.

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

%Y Cf. A000720, A076610, A340852, A344606, A355737, A355739, A355740, A355741, A355744, A355747.

%K nonn

%O 1,2

%A _Gus Wiseman_, Jul 21 2022