A368680 Sum of the numbers k less than n and not dividing n such that n-k is squarefree.
0, 0, 2, 3, 9, 9, 17, 21, 27, 31, 42, 42, 59, 60, 72, 86, 100, 108, 124, 121, 143, 152, 178, 189, 211, 214, 243, 238, 278, 280, 314, 332, 341, 358, 392, 409, 444, 448, 479, 501, 545, 540, 599, 593, 640, 664, 716, 739, 772, 810, 824, 833, 905, 934, 971, 990, 1020
Offset: 1
Examples
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.
Programs
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Mathematica
Table[Sum[k * MoebiusMu[n - k]^2 (Ceiling[n/k] - Floor[n/k]), {k, n}], {n, 100}] -
PARI
a(n) = sum(k=1, n-1, if ((n % k) && issquarefree(n-k), k)); \\ Michel Marcus, Jan 03 2024
Formula
a(n) = Sum_{k=1..n} k * mu(n-k)^2 * (ceiling(n/k) - floor(n/k)).