A291574 The arithmetic function uhat(n,6,6).

1, 0, -1, 0, 1, -4, 1, 0, -1, 0, 1, -4, 1, 0, -1, 0, 1, -4, 1, 0, -1, 0, 1, -4, 1, 0, -1, 0, 1, -4, 1, 0, -1, 0, 1, -4, 1, 0, -1, 0, 1, -4, 1, 0, -1, 0, 1, -4, 1, 0, -1, 0, 1, -4, 1, 0, -1, 0, 1, -4, 1, 0, -1, 0, 1, -4, 1, 0, -1, 0

Offset: 1

Author

Robert Price, Aug 26 2017

Keywords

• sign

Links

• Bela Bajnok, Additive Combinatorics: A Menu of Research Problems, arXiv:1705.07444 [math.NT], May 2017. See Table in Section 1.6.4.

Crossrefs

Cf. A289435, A289436, A289437, A289438, A289439, A289440, A289441.

Programs

Mathematica

delta[r_, k_, d_] := If[r < k, (k - r)*r - (d - 1), If[k < r && r < d, (d - r)*(r - k) - (d - 1), If[k == r && r == d, d - 1, 0]]] uhat[n_, m_, h_] := (dx = Divisors[n]; dmin = n; For[i = 1, i ≤ Length[dx], i++, d = dx[[i]]; k = m - d*Ceiling[m/d] + d; r = h - d*Ceiling[h/d] + d; If[h ≤ Min[k, d - 1], dmin = Min[dmin, n, (h*Ceiling[m/d] - h + 1)*d, h*m - h*h + 1], dmin = Min[dmin, n, h*m - h*h + 1 - delta[r, k, d]]]]; dmin) Table[uhat[n, 6, 6], {n, 1, 70}]

Formula

Conjectures from Chai Wah Wu, Jun 10 2025: (Start)

a(n) = a(n-6) for n > 6.

G.f.: x*(4*x^5 - x^4 + x^2 - 1)/(x^6 - 1). (End)