OFFSET
1,5
COMMENTS
In this sequence we consider the vector space of real numbers (R) with scalar multiplication by rational numbers (Q).
For any n > 0:
- the linear combinations of elements of { log(1), ..., log(n) }, say V_n, constitute a subspace with dimension PrimePi(n),
- (log(2), log(3), ..., log(prime(PrimePi(n)))) is a base of V_n,
- A307984(n) gives the numbers of bases of V_n.
LINKS
Rémy Sigrist, PARI program for A307998
FORMULA
EXAMPLE
The triangle begins:
n\k| 0 1 2 3 4 5
---+-----------------------
1| 1
2| 1 1
3| 1 2 1
4| 1 3 2
5| 1 4 5 2
6| 1 5 9 5
7| 1 6 14 14 5
8| 1 7 18 19 7
9| 1 8 24 28 11
10| 1 9 32 49 25
11| 1 10 41 81 74 25
...
For n = 4:
- T(4, 0) = #{ {} } = 1,
- T(4, 1) = #{ {log(2)}, {log(3)}, {log(4)} } = 3,
- T(4, 2) = #{ {log(2), log(3)}, {log(3), log(4)} } = 2,
- log(2) = log(4)/2, so log(2) and log(4)} are Q-linearly dependent.
PROG
(PARI) See Links section.
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Rémy Sigrist, May 09 2019
STATUS
approved