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

1, 3, 7, 13, 21, 35, 39, 89, 133, 105, 91, 195, 351, 285, 247, 333, 273, 481, 455, 555, 623, 801, 791, 741, 1359, 1157, 1281, 1335, 1365, 1443, 1977, 1729, 1967, 1869, 2109, 3185, 2373, 2769, 2639, 4361, 3367, 3653, 3885, 3471, 4613, 5883, 5187, 5551, 6327

Offset

1,2

Comments

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

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.

Links

Table of n, a(n) for n=1..49.

Example

The terms together with their prime indices begin:
    1: {}
    3: {2}
    7: {4}
   13: {6}
   21: {2,4}
   35: {3,4}
   39: {2,6}
   89: {24}
  133: {4,8}
  105: {2,3,4}
   91: {4,6}
  195: {2,3,6}
  351: {2,2,2,6}
For example, the choices for a(12) = 195 are:
  {1,1,1}  {1,2,2}  {1,3,6}
  {1,1,2}  {1,2,3}  {2,2,3}
  {1,1,3}  {1,2,6}  {2,3,3}
  {1,1,6}  {1,3,3}  {2,3,6}

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
mnrm[s_]:=If[Min@@s==1, mnrm[DeleteCases[s-1, 0]]+1, 0];
az=Table[Length[Union[Sort/@Tuples[Divisors/@primeMS[n]]]], {n, 1000}];
Table[Position[az, k][[1, 1]], {k, mnrm[az]}]

Crossrefs

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

Positions of first appearances in A355733.

Choosing weakly increasing divisors gives A355736, firsts of A355735.

A000005 counts divisors.

A001414 adds up distinct prime divisors, counted by A001221.

A003963 multiplies together the prime indices of n.

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

A120383 lists numbers divisible by all of their prime indices.

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

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

Sequence in context: A357370 A060939 A174030 * A098575 A363144 A138035

Adjacent sequences: A355731 A355732 A355733 * A355735 A355736 A355737

Keyword

nonn

Author

Gus Wiseman, Jul 21 2022

Status

approved