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

-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, 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, 20, 1, 8, 44, 24, 1, 1, 1, 18, 8, 9, 60, 8

Offset: 2

Author

Mats Granvik, Apr 19 2021

Keywords

• frac
• sign

Examples

			max(-1/2) = -1/2 therefore a(2) = -1,
max(1/2, -2/3) = 1/2 therefore a(3) = 1,
max(-1/2, 1/3, -1/4) = 1/3 therefore a(4) = 1,
max(1/2, 1/3, 1/4, -4/5) = 1/2 therefore a(5) = 1
max(-1/2, -2/3, -1/4, 1/5, 1/3) = 1/3 therefore a(6) = 1,
max(1/2, 1/3, 1/4, 1/5, 1/6, -6/7) = 1/2 therefore a(7) = 1,
max(-1/2, 1/3, -1/4, 1/5, -1/6, 1/7, -1/8) = 1/3 therefore a(8) = 1,
max(1/2, -2/3, 1/4, 1/5, -1/3, 1/7, 1/8, -2/9) = 1/2 therefore a(9) = 1,
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.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 2..20000

Crossrefs

Cf. A343556 (denominator). Cf. A171462, A191898.

Programs

Mathematica

a[n_] := DivisorSum[n, MoebiusMu[#] # &]; nn = 78; Numerator[
Table[Max[Table[a[GCD[n, k]]/k, {k, 2, n}]], {n, 2, nn}]]

PARI

memoA191898 = Map();
A191898sq(n, k) = if(n<1 || k<1, 0, n==1 || k==1, 1, k>n, A191898sq(k, n), kA191898sq(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
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

Formula

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