A343556 a(n) = denominator(max_{k=2..n}(A191898(n, k)/k)), n>=2.

2, 2, 3, 2, 3, 2, 3, 2, 5, 2, 3, 2, 7, 15, 3, 2, 3, 2, 5, 7, 11, 2, 3, 2, 13, 2, 7, 2, 15, 2, 3, 33, 17, 35, 3, 2, 19, 13, 5, 2, 7, 2, 11, 15, 23, 2, 3, 2, 5, 51, 13, 2, 3, 11, 7, 19, 29, 2, 15, 2, 31, 7, 3, 65, 33, 2, 17, 69, 35, 2, 3, 2, 37, 15, 19, 77, 13

Offset: 2

Author

Mats Granvik, Apr 19 2021

Keywords

• nonn
• frac

Examples

			max(-1/2) = -1/2 therefore a(2) = 2,
max(1/2, -2/3) = 1/2 therefore a(3) = 2,
max(-1/2, 1/3, -1/4) = 1/3 therefore a(4) = 3,
max(1/2, 1/3, 1/4, -4/5) = 1/2 therefore a(5) = 2
max(-1/2, -2/3, -1/4, 1/5, 1/3) = 1/3 therefore a(6) = 3,
max(1/2, 1/3, 1/4, 1/5, 1/6, -6/7) = 1/2 therefore a(7) = 2,
max(-1/2, 1/3, -1/4, 1/5, -1/6, 1/7, -1/8) = 1/3 therefore a(8) = 3,
max(1/2, -2/3, 1/4, 1/5, -1/3, 1/7, 1/8, -2/9) = 1/2 therefore a(9) = 2,
max(-1/2, 1/3, -1/4, -4/5, -1/6, 1/7, -1/8, 1/9, 2/5) = 2/5 therefore a(10) = 5.
		

Links

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

Crossrefs

Cf. A343555 (numerators). Cf. A171462, A191898.

Programs

Mathematica

a[n_] := DivisorSum[n, MoebiusMu[#] # &]; nn = 78; Denominator[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
A343556(n) = { my(m=0,r); for(k=2, n, r = A191898sq(n, k)/k; if(!m || (r > m), m = r)); denominator(m); }; \\ Antti Karttunen, Jan 28 2025

Formula

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