OFFSET
1,12
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.
We define a semi-sum of a multiset to be any sum of a 2-element submultiset. This is different from sums of pairs of elements. For example, 2 is the sum of a pair of elements of {1}, but there are no semi-sums.
EXAMPLE
The prime indices of 90 are {1,2,2,3}, with semi-sums
3 = 1+2
4 = 1+3 (or 2+2)
5 = 2+3
so a(90) = 3.
Alternatively, the semiprime divisors of 90 are (6,9,10,15), with prime indices ({1,2},{2,2},{1,3},{2,3}) with sums (3,4,4,5) so a(90) = 3.
MATHEMATICA
prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Table[Length[Union[Total/@Subsets[prix[n], {2}]]], {n, 100}]
CROSSREFS
The non-binary version is A299701.
For all sums of pairs of elements we have A367095.
Positions of first appearances are A367097.
Semiprime divisors are listed by A367096 and have:
- square count: A056170
- sum: A076290
- squarefree count: A079275
- count: A086971
- firsts: A220264
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 04 2023
STATUS
approved