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

1,3

Links

Table of n, a(n) for n=1..57.

Formula

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

Example

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.

Mathematica

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

Prog

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

Crossrefs

Cf. A008683 (mu), A368677, A368679.

Sequence in context: A337998 A131988 A092593 * A231365 A024816 A196443

Adjacent sequences: A368677 A368678 A368679 * A368681 A368682 A368683

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Jan 02 2024

Status

approved