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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 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

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