A307998 - OEIS

%I #15 Dec 09 2024 05:49:55

%S 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,

%T 11,1,9,32,49,25,1,10,41,81,74,25,1,11,51,111,108,38,1,12,62,162,219,

%U 146,38,1,13,74,221,351,276,84,1,14,87,293,526,457,150

%N 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).

%C In this sequence we consider the vector space of real numbers (R) with scalar multiplication by rational numbers (Q).

%C For any n > 0:

%C - the linear combinations of elements of { log(1), ..., log(n) }, say V_n, constitute a subspace with dimension PrimePi(n),

%C - (log(2), log(3), ..., log(prime(PrimePi(n)))) is a base of V_n,

%C - A307984(n) gives the numbers of bases of V_n.

%H Rémy Sigrist, <a href="/A307998/a307998.gp.txt">PARI program for A307998</a>

%F T(n, 0) = 1 for any n > 0.

%F T(n, 1) = n-1 for any n > 1.

%F T(n, A000720(n)) = A307984(n) for any n > 0.

%F T(p, k) = T(p-1, k-1) + T(p-1, k) for the n-th prime number p and k = 1..n-1.

%e The triangle begins:

%e   n\k|  0   1   2   3   4   5

%e   ---+-----------------------

%e     1|  1

%e     2|  1   1

%e     3|  1   2   1

%e     4|  1   3   2

%e     5|  1   4   5   2

%e     6|  1   5   9   5

%e     7|  1   6  14  14   5

%e     8|  1   7  18  19   7

%e     9|  1   8  24  28  11

%e    10|  1   9  32  49  25

%e    11|  1  10  41  81  74  25

%e    ...

%e For n = 4:

%e - T(4, 0) = #{ {} } = 1,

%e - T(4, 1) = #{ {log(2)}, {log(3)}, {log(4)} } = 3,

%e - T(4, 2) = #{ {log(2), log(3)}, {log(3), log(4)} } = 2,

%e - log(2) = log(4)/2, so log(2) and log(4) are Q-linearly dependent.

%o (PARI) See Links section.

%Y Cf. A000720, A307984.

%K nonn,tabf

%O 1,5

%A _Rémy Sigrist_, May 09 2019