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A307984 a(n) is the number of Q-bases which can be built from the set {log(1),...,log(n)}.

%I A307984 #52 Aug 03 2024 17:56:33
%S A307984 1,1,1,2,2,5,5,7,11,25,25,38,38,84,150,178,178,235,235,341,578,1233,
%T A307984 1233,1521,1966,4156,4820,6832,6832,8952,8952,9824,15926,33256,47732,
%U A307984 54488,54488,113388,181728,218592,218592,279348,279348,388576,467028,966700,966700
%N A307984 a(n) is the number of Q-bases which can be built from the set {log(1),...,log(n)}.
%C A307984 The real numbers log(p_1),...,log(p_r) where p_i is the i-th prime are known to be linearly independent over the rationals Q. Hence, for the numbers {log(1),...,log(n)}, where pi(n) = r, those numbers log(p_i) form a Q-basis of V_n:= <log(1),...,log(n)> = the Q-vector space generated by {log(1),...,log(n)}. This sequence a(n) counts the different Q-bases of V_n which can be build from the vectors of the set {log(1),...,log(n)}.
%C A307984 First differs from A370585 at A370585(21) = 579, a(21) = 578. The difference is due to the set {10,11,13,14,15,17,19,21}, which is not a basis because log(10) + log(21) = log(14) + log(15). - _Gus Wiseman_, Mar 13 2024
%H A307984 MathOverflow, related: <a href="https://mathoverflow.net/questions/325880/is-this-line-of-thought-using-linear-algebra-to-get-number-theoretic-results-a">'Linear Algebra in Number Theory'</a>
%F A307984 a(p) = a(p-1) for any prime number p. - _Rémy Sigrist_, May 09 2019
%e A307984 [{}] -> For n = 1, we have 1 = a(1) bases; we count {} as a basis for V_0 = {0};
%e A307984 [{2}] -> for n = 2, we have 1 = a(2) basis, which is {2};
%e A307984 [{2, 3}] -> for n = 3, we have 1 = a(3) basis, which is {2,3};
%e A307984 [{2, 3}, {3, 4}] -> for n = 4 we have 2 = a(4) bases, which are {2,3},{3,4};
%e A307984 [{2, 3, 5}, {3, 4, 5}] -> a(5) = 2;
%e A307984 [{2, 3, 5}, {2, 5, 6}, {3, 4, 5}, {3, 5, 6}, {4, 5, 6}] -> a(6) = 5;
%e A307984 [{2, 3, 5, 7}, {2, 5, 6, 7}, {3, 4, 5, 7}, {3, 5, 6, 7}, {4, 5, 6, 7}] -> a(7) = 5.
%o A307984 (Sage)
%o A307984 MAXN=100
%o A307984 def Log(a,N=MAXN):
%o A307984     return vector([valuation(a,p) for p in primes(N)])
%o A307984 def allBases(n,N=MAXN):
%o A307984     M = matrix([Log(n,N=N) for n in range(1,n+1)],ring=QQ)
%o A307984     r = M.rank()
%o A307984     rr = Set(range(1,n+1))
%o A307984     ll = []
%o A307984     for S in rr.subsets(r):
%o A307984         M = matrix([Log(k,N=N) for k in S])
%o A307984         if M.rank()==r:
%o A307984             ll.append(S)
%o A307984     return ll
%o A307984 [len(allBases(k)) for k in range(1,12)]
%Y A307984 A370585 counts maximal factor-choosable subsets.
%Y A307984 Cf. A333331, A355529, A370582, A370640.
%K A307984 nonn,nice
%O A307984 1,4
%A A307984 _Orges Leka_, May 09 2019
%E A307984 a(12)-a(47) from _Rémy Sigrist_, May 09 2019