A385215 Number of maximal sparse submultisets of the prime indices of n, where a multiset is sparse iff 1 is not a first difference.

1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1

Offset

1,6

Comments

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..87.

Formula

a(n) <= A166469(n).

Example

The sparse submultisets of the prime indices of n = 8 are {{},{1},{1,1},{1,1,1}}, with maximization {{1,1,1}}. So a(8) = 1.
The sparse submultisets of the prime indices of n = 462 are {{},{1},{2},{4},{5},{1,4},{2,4},{1,5},{2,5}}, with maximization {{1,4},{1,5},{2,4},{2,5}}, so a(462) = 4.
The prime indices of n together their a(n) maximal sparse submultisets for n = 1, 6, 210, 462, 30030, 46410:
  {}  {1,2}  {1,2,3,4}  {1,2,4,5}  {1,2,3,4,5,6}  {1,2,3,4,6,7}
  ------------------------------------------------------------
  {}   {1}     {1,3}      {1,4}       {2,5}          {1,3,6}
       {2}     {1,4}      {1,5}       {1,3,5}        {1,3,7}
               {2,4}      {2,4}       {1,3,6}        {1,4,6}
                          {2,5}       {1,4,6}        {1,4,7}
                                      {2,4,6}        {2,4,6}
                                                     {2,4,7}

Mathematica

prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
maxq[els_]:=Select[els, Not[Or@@Table[Divisible[oth, #], {oth, DeleteCases[els, #]}]]&];
Table[Length[maxq[Select[Divisors[n], FreeQ[Differences[prix[#]], 1]&]]], {n, 30}]

Crossrefs

This is the maximal case of A166469.

For binary instead of prime indices we have A384883, maximal case of A245564.

The greatest number whose prime indices are one of these submultisets is A385216.

A034839 counts subsets by number of maximal runs, for strict partitions A116674.

A384887 counts partitions with equal lengths of gapless runs, distinct A384884.

A384893 counts subsets by number of maximal anti-runs, for partitions A268193, A384905.

Cf. A000045, A000071, A001629, A202064, A316476, A319630, A374356, A384890.

Sequence in context: A374586 A280801 A085006 * A378886 A384906 A300820

Adjacent sequences: A385212 A385213 A385214 * A385216 A385217 A385218

Keyword

nonn

Author

Gus Wiseman, Jul 03 2025

Status

approved