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A307998 Irregular triangle read by rows, n > 0 and k = 0..PrimePi(n): T(n, k) is the number of Q-linearly independent subsets of { log(1), ..., log(n) } with k elements (where PrimePi corresponds to A000720, the prime-counting function). 1
1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 4, 5, 2, 1, 5, 9, 5, 1, 6, 14, 14, 5, 1, 7, 18, 19, 7, 1, 8, 24, 28, 11, 1, 9, 32, 49, 25, 1, 10, 41, 81, 74, 25, 1, 11, 51, 111, 108, 38, 1, 12, 62, 162, 219, 146, 38, 1, 13, 74, 221, 351, 276, 84, 1, 14, 87, 293, 526, 457, 150 (list; graph; refs; listen; history; text; internal format)
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
FORMULA
T(n, 0) = 1 for any n > 0.
T(n, 1) = n-1 for any n > 1.
T(n, A000720(n)) = A307984(n) for any n > 0.
T(p, k) = T(p-1, k-1) + T(p-1, k) for the n-th prime number p and k = 1..n-1.
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
Sequence in context: A089177 A348445 A023996 * A355147 A049998 A029253
KEYWORD
nonn,tabf
AUTHOR
Rémy Sigrist, May 09 2019
STATUS
approved

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Last modified May 22 04:05 EDT 2024. Contains 372742 sequences. (Running on oeis4.)