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

Author

Rémy Sigrist, May 09 2019

Keywords

• nonn
• tabf

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.

Examples

			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.
		

Links

• Rémy Sigrist, PARI program for A307998

Crossrefs

Cf. A000720, A307984.

Programs

PARI

See Links section.

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.