A382775 Least number appearing n times in A048767 (rank of Look-and-Say partition of prime indices).

6, 1, 8, 32, 64, 128, 256, 6144, 512, 27648, 1024, 73728, 2048, 147456, 165888, 4096, 248832, 196608, 8192, 497664, 1119744, 393216, 16384, 2239488

Offset: 0

Author

Gus Wiseman, Apr 11 2025

Keywords

• nonn
• more

Comments

Also the position of first appearance of n in A382525 (number of times n appears in A048767).

The Look-and-Say partition of a multiset or partition y is obtained by interchanging parts with multiplicities. Hence, the multiplicity of k in the Look-and-Say partition of y is the sum of all parts that appear exactly k times. For example, starting with (3,2,2,1,1) we get (2,2,2,1,1,1), the multiset union of ((1,1,1),(2,2),(2)).

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, sum A056239.

Examples

			The terms together with their prime indices begin:
       6: {1,2}
       1: {}
       8: {1,1,1}
      32: {1,1,1,1,1}
      64: {1,1,1,1,1,1}
     128: {1,1,1,1,1,1,1}
     256: {1,1,1,1,1,1,1,1}
    6144: {1,1,1,1,1,1,1,1,1,1,1,2}
     512: {1,1,1,1,1,1,1,1,1}
   27648: {1,1,1,1,1,1,1,1,1,1,2,2,2}
    1024: {1,1,1,1,1,1,1,1,1,1}
   73728: {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
    2048: {1,1,1,1,1,1,1,1,1,1,1}
  147456: {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
  165888: {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
    4096: {1,1,1,1,1,1,1,1,1,1,1,1}
  248832: {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
		

Crossrefs

Positions of first appearances in A382525.

The Look-and-Say partition is ranked by A048767, listed by A381440.

Look-and-Say partitions are counted by A239455, complement A351293.

Look-and-Say partitions are ranked by A351294.

Non-Look-and-Say partitions are ranked by A351295, conjugate A381433.

The section-sum partition is ranked by A381431, listed by A381436.

Section-sum partitions are ranked by A381432.

A055396 gives least prime index, greatest A061395.

A056239 adds up prime indices, row sums of A112798.

A122111 represents conjugation in terms of Heinz numbers.

Cf. A003557, A047966, A048768, A051903, A051904, A066328, A071178, A116861, A130091, A217605, A239964, A381540.

Programs

Mathematica

stp[y_]:=Select[Tuples[Select[IntegerPartitions[#], UnsameQ@@#&]&/@y],UnsameQ@@Join@@#&];
z=Table[Length[stp[Last/@FactorInteger[n]]],{n,10000}];
mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
Table[Position[z,k][[1,1]],{k,0,mnrm[z+1]-1}]