OFFSET
1,1
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)).
EXAMPLE
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}
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&]
CROSSREFS
KEYWORD
nonn,more,new
AUTHOR
Gus Wiseman, Mar 02 2025
STATUS
approved