A355732 Least k such that there are exactly n ways to choose a sequence of divisors, one of each element of the multiset of prime indices of k (with multiplicity).

1, 3, 7, 9, 53, 21, 311, 27, 49, 159, 8161, 63, 38873, 933, 371, 81, 147, 477, 2177, 24483, 189, 2809, 343, 2799, 1113, 243, 57127, 16483, 441, 1431, 6531, 73449, 2597, 567, 96721, 8427, 1029, 8397, 3339, 15239, 729, 49449, 1323, 19663, 4293, 2401, 19593, 7791

Offset

1,2

Comments

This is the position of first appearance of n in A355731.

Appears to be a subset of A353397.

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..48.

Example

The terms together with their prime indices begin:
      1: {}
      3: {2}
      7: {4}
      9: {2,2}
     53: {16}
     21: {2,4}
    311: {64}
     27: {2,2,2}
     49: {4,4}
    159: {2,16}
   8161: {1024}
     63: {2,2,4}
For example, the choices for a(12) = 63 are:
  (1,1,1)  (1,2,2)  (2,1,4)
  (1,1,2)  (1,2,4)  (2,2,1)
  (1,1,4)  (2,1,1)  (2,2,2)
  (1,2,1)  (2,1,2)  (2,2,4)

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[Times@@Length/@Divisors/@primeMS[n], {n, 1000}];
Table[Position[az, k][[1, 1]], {k, mnrm[az]}]

Crossrefs

Positions of first appearances in A355731.

Counting distinct sequences after sorting: A355734, firsts of A355733.

Requiring the result to be weakly increasing: A355736, firsts of A355735.

Requiring the result to be relatively prime: A355738, firsts of A355737.

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, A340606, A355739, A355740, A355741, A355742, A355744, A355746, A355747, A355748.

Sequence in context: A128052 A033681 A074339 * A115164 A088801 A003033

Adjacent sequences: A355729 A355730 A355731 * A355733 A355734 A355735

Keyword

nonn

Author

Gus Wiseman, Jul 21 2022

Status

approved