A368679 Sum of the squarefree values of (n-k) where the numbers k are the numbers less than n that do not divide n.

0, 0, 1, 1, 6, 3, 11, 11, 18, 19, 24, 18, 45, 38, 48, 58, 87, 72, 104, 79, 109, 112, 144, 123, 189, 176, 189, 154, 215, 200, 244, 244, 253, 288, 308, 275, 407, 388, 418, 379, 521, 426, 562, 507, 575, 624, 647, 605, 698, 740, 706, 675, 791, 740, 844, 802, 861, 870, 956

Offset: 1

Author

Wesley Ivan Hurt, Jan 02 2024

Keywords

• nonn
• easy

Examples

			a(12) = 18. The numbers k that are less than 12 and do not divide 12 are: {5,7,8,9,10,11}. The corresponding n-k values are: {7,5,4,3,2,1} (only 5 of which are squarefree). The sum of the squarefree values of n-k is then 7+5+3+2+1 = 18.
		

Crossrefs

Cf. A008683 (mu), A368677, A368680.

Programs

Mathematica

Table[Sum[(n - 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), n-k)); \\ Michel Marcus, Jan 03 2024

Formula

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