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Sum of the numbers k less than n and not dividing n such that n-k is squarefree.
2

%I #9 Jan 03 2024 05:33:57

%S 0,0,2,3,9,9,17,21,27,31,42,42,59,60,72,86,100,108,124,121,143,152,

%T 178,189,211,214,243,238,278,280,314,332,341,358,392,409,444,448,479,

%U 501,545,540,599,593,640,664,716,739,772,810,824,833,905,934,971,990,1020

%N Sum of the numbers k less than n and not dividing n such that n-k is squarefree.

%F a(n) = Sum_{k=1..n} k * mu(n-k)^2 * (ceiling(n/k) - floor(n/k)).

%e a(12) = 42. The numbers less than 12 that do not divide 12 are: {5,7,8,9,10,11} with corresponding values of n-k: {7,5,4,3,2,1} (all of which are squarefree, except 4). Adding the values of k that give squarefree n-k, we have: 5+7+9+10+11 = 42.

%t Table[Sum[k * MoebiusMu[n - k]^2 (Ceiling[n/k] - Floor[n/k]), {k, n}], {n, 100}]

%o (PARI) a(n) = sum(k=1, n-1, if ((n % k) && issquarefree(n-k), k)); \\ _Michel Marcus_, Jan 03 2024

%Y Cf. A008683 (mu), A368677, A368679.

%K nonn,easy

%O 1,3

%A _Wesley Ivan Hurt_, Jan 02 2024