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, 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

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

Table of n, a(n) for n=1..70.

Rémy Sigrist, PARI program for A307998

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

Cf. A000720, A307984.

Sequence in context: A089177 A348445 A023996 * A355147 A049998 A029253

Adjacent sequences: A307995 A307996 A307997 * A307999 A308000 A308001

Keyword

nonn,tabf

Author

Rémy Sigrist, May 09 2019

Status

approved