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A343555 a(n) = numerator(max_{k=2..n}(A191898(n, k)/k)), n>=2.

%I A343555 #18 Jan 28 2025 15:30:08
%S A343555 -1,1,1,1,1,1,1,1,2,1,1,1,3,8,1,1,1,1,2,4,5,1,1,1,6,1,3,1,8,1,1,20,8,
%T A343555 24,1,1,9,8,2,1,4,1,5,8,11,1,1,1,2,32,6,1,1,8,3,12,14,1,8,1,15,4,1,48,
%U A343555 20,1,8,44,24,1,1,1,18,8,9,60,8
%N A343555 a(n) = numerator(max_{k=2..n}(A191898(n, k)/k)), n>=2.
%H A343555 Antti Karttunen, <a href="/A343555/b343555.txt">Table of n, a(n) for n = 2..20000</a>
%F A343555 n>=2: a(n) = numerator(max_{k=2..n}(A191898(n, k)/k)).
%e A343555 max(-1/2) = -1/2 therefore a(2) = -1,
%e A343555 max(1/2, -2/3) = 1/2 therefore a(3) = 1,
%e A343555 max(-1/2, 1/3, -1/4) = 1/3 therefore a(4) = 1,
%e A343555 max(1/2, 1/3, 1/4, -4/5) = 1/2 therefore a(5) = 1
%e A343555 max(-1/2, -2/3, -1/4, 1/5, 1/3) = 1/3 therefore a(6) = 1,
%e A343555 max(1/2, 1/3, 1/4, 1/5, 1/6, -6/7) = 1/2 therefore a(7) = 1,
%e A343555 max(-1/2, 1/3, -1/4, 1/5, -1/6, 1/7, -1/8) = 1/3 therefore a(8) = 1,
%e A343555 max(1/2, -2/3, 1/4, 1/5, -1/3, 1/7, 1/8, -2/9) = 1/2 therefore a(9) = 1,
%e A343555 max(-1/2, 1/3, -1/4, -4/5, -1/6, 1/7, -1/8, 1/9, 2/5) = 2/5 therefore a(10) = 2.
%t A343555 a[n_] := DivisorSum[n, MoebiusMu[#] # &]; nn = 78; Numerator[
%t A343555 Table[Max[Table[a[GCD[n, k]]/k, {k, 2, n}]], {n, 2, nn}]]
%o A343555 (PARI)
%o A343555 memoA191898 = Map();
%o A343555 A191898sq(n, k) = if(n<1 || k<1, 0, n==1 || k==1, 1, k>n, A191898sq(k, n), k<n, A191898sq(k, (n-1)%k+1), my(v); if(mapisdefined(memoA191898,[n,k],&v), v, v = -sum( i=1, n-1, A191898sq(n, i)); mapput(memoA191898,[n,k],v); (v))); \\ After _Michael Somos_' code in A191898
%o A343555 A343555(n) = { my(m=0,r); for(k=2, n, r = A191898sq(n, k)/k; if(!m || (r > m), m = r)); numerator(m); }; \\ _Antti Karttunen_, Jan 28 2025
%Y A343555 Cf. A343556 (denominator). Cf. A171462, A191898.
%K A343555 frac,sign
%O A343555 2,9
%A A343555 _Mats Granvik_, Apr 19 2021