A381541 Numbers appearing more than once in A048767 (Look-and-Say partition of prime indices).

8, 16, 27, 32, 64, 81, 96, 125, 128, 144, 160, 192, 216, 224, 243, 256, 288

Offset: 1

Author

Gus Wiseman, Mar 02 2025

Keywords

• nonn
• more

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.

The Look-and-Say partition of a multiset or partition y is obtained by interchanging parts with multiplicities. 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)).

The conjugate of a Look-and-Say partition is a section-sum partition; see A381431, union A381432, count A239455.

Examples

			The terms together with their prime indices begin:
    8: {1,1,1}
   16: {1,1,1,1}
   27: {2,2,2}
   32: {1,1,1,1,1}
   64: {1,1,1,1,1,1}
   81: {2,2,2,2}
   96: {1,1,1,1,1,2}
  125: {3,3,3}
  128: {1,1,1,1,1,1,1}
  144: {1,1,1,1,2,2}
  160: {1,1,1,1,1,3}
  192: {1,1,1,1,1,1,2}
  216: {1,1,1,2,2,2}
  224: {1,1,1,1,1,4}
  243: {2,2,2,2,2}
  256: {1,1,1,1,1,1,1,1}
  288: {1,1,1,1,1,2,2}
For example, the term 96 appears in A048767 at positions 44 and 60, with prime indices:
  44: {1,1,5}
  60: {1,1,2,3}
		

Crossrefs

In A048767:

- fixed points are A048768, A217605

- conjugate is A381431, fixed points A000961, A000005

- all numbers present are A351294, conjugate A381432

- numbers missing are A351295, conjugate A381433

- numbers appearing only once are A381540, conjugate A381434

- numbers appearing more than once are A381541 (this), conjugate A381435

A000040 lists the primes.

A055396 gives least prime index, greatest A061395.

A056239 adds up prime indices, row sums of A112798.

A122111 represents conjugation in terms of Heinz numbers.

A239455 counts Look-and-Say partitions, complement A351293.

A381440 lists Look-and-Say partitions of prime indices, conjugate A381436.

Cf. A000720, A001222, A003557, A047966, A051903, A051904, A066328, A071178, A116861, A130091, A239964.

Programs

Mathematica

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
hls[y_]:=Product[Prime[Count[y,x]]^x,{x,Union[y]}];
Select[Range[100],Count[hls/@IntegerPartitions[Total[prix[#]]],#]>1&]