This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A307998 #15 Dec 09 2024 05:49:55
%S A307998 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 A307998 11,1,9,32,49,25,1,10,41,81,74,25,1,11,51,111,108,38,1,12,62,162,219,
%U A307998 146,38,1,13,74,221,351,276,84,1,14,87,293,526,457,150
%N 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).
%C A307998 In this sequence we consider the vector space of real numbers (R) with scalar multiplication by rational numbers (Q).
%C A307998 For any n > 0:
%C A307998 - the linear combinations of elements of { log(1), ..., log(n) }, say V_n, constitute a subspace with dimension PrimePi(n),
%C A307998 - (log(2), log(3), ..., log(prime(PrimePi(n)))) is a base of V_n,
%C A307998 - A307984(n) gives the numbers of bases of V_n.
%H A307998 Rémy Sigrist, <a href="/A307998/a307998.gp.txt">PARI program for A307998</a>
%F A307998 T(n, 0) = 1 for any n > 0.
%F A307998 T(n, 1) = n-1 for any n > 1.
%F A307998 T(n, A000720(n)) = A307984(n) for any n > 0.
%F A307998 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 A307998 The triangle begins:
%e A307998 n\k| 0 1 2 3 4 5
%e A307998 ---+-----------------------
%e A307998 1| 1
%e A307998 2| 1 1
%e A307998 3| 1 2 1
%e A307998 4| 1 3 2
%e A307998 5| 1 4 5 2
%e A307998 6| 1 5 9 5
%e A307998 7| 1 6 14 14 5
%e A307998 8| 1 7 18 19 7
%e A307998 9| 1 8 24 28 11
%e A307998 10| 1 9 32 49 25
%e A307998 11| 1 10 41 81 74 25
%e A307998 ...
%e A307998 For n = 4:
%e A307998 - T(4, 0) = #{ {} } = 1,
%e A307998 - T(4, 1) = #{ {log(2)}, {log(3)}, {log(4)} } = 3,
%e A307998 - T(4, 2) = #{ {log(2), log(3)}, {log(3), log(4)} } = 2,
%e A307998 - log(2) = log(4)/2, so log(2) and log(4) are Q-linearly dependent.
%o A307998 (PARI) See Links section.
%Y A307998 Cf. A000720, A307984.
%K A307998 nonn,tabf
%O A307998 1,5
%A A307998 _Rémy Sigrist_, May 09 2019