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A366739
Number of distinct semi-sums of the multiset of prime indices of n. Number of distinct sums of prime indices of semiprime divisors of n (counted by A086971).
11
0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 2, 1, 1, 0, 2, 1, 1, 1, 2, 0, 3, 0, 1, 1, 1, 1, 3, 0, 1, 1, 2, 0, 3, 0, 2, 2, 1, 0, 2, 1, 2, 1, 2, 0, 2, 1, 2, 1, 1, 0, 4, 0, 1, 2, 1, 1, 3, 0, 2, 1, 3, 0, 3, 0, 1, 2, 2, 1, 3, 0, 2, 1, 1, 0, 4, 1, 1, 1, 2, 0, 3
OFFSET
1,12
COMMENTS
First differs from A086971 at a(90) = 3, A086971(90) = 4.
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.
Summing over partitions gives A366738, strict A366741.
For all sums of pairs of elements we have A367095.
Positions of first appearances are A367097.
A001222 counts prime factors (or prime indices), distinct A001221.
A001358 lists semiprimes, squarefree A006881, conjugate A065119.
A056239 adds up prime indices, row sums of A112798.
A299702 ranks knapsack partitions, counted by A108917.
Semiprime divisors are listed by A367096 and have:
- square count: A056170
- sum: A076290
- squarefree count: A079275
- count: A086971
- firsts: A220264
Sequence in context: A294881 A355523 A110475 * A086971 A341677 A211159
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 04 2023
STATUS
approved