The number of numbers i such that p = i (mod n) for no powerful number p and 0 <= i < n.
0, 0, 0, 1, 0, 0, 0, 2, 2, 0, 0, 3, 0, 0, 0, 4, 0, 4, 0, 5, 0, 0, 0, 6, 4, 0, 6, 7, 0, 0, 0, 8, 0, 0, 0, 15, 0, 0, 0, 10, 0, 0, 0, 11, 10, 0, 0, 12, 6, 8, 0, 13, 0, 12, 0, 14, 0, 0, 0, 15, 0, 0, 14, 16, 0, 0, 0, 17, 0, 0, 0, 30, 0, 0, 12, 19, 0, 0, 0, 20, 18, 0
1
The number is zero when n is squarefree.
T. D. Noe, Plot of 200 terms
T. D. Noe, Table of 200 terms
(Mma) nn = 8; nMax = 10^nn; lst = {}; Do[lst = Join[lst, i^3 Range[Sqrt[nMax/i^3]]^2], {i, nMax^(1/3)}]; Table[Length[Complement[Range[0, n - 1], Transpose[Tally[Mod[lst, n]]][[1]]]], {n, 200}]
Cf. A001694 (powerful), A005117 (squarefree), A013929 (not squarefree), S000119, S000121.
nonn
T. D. Noe, Jul 03 2014