A291515 The arithmetic function uhat(n,3,6).
-17, -17, -19, -17, -17, -19, -17, -17, -19, -20, -22, -24, -26, -28, -30, -32, -34, -36, -38, -40, -42, -44, -46, -48, -50, -52, -54, -56, -58, -60, -62, -64, -66, -68, -70, -72, -74, -76, -78, -80, -82, -84, -86, -88, -90, -92, -94, -96, -98, -100, -102, -104, -106, -108, -110, -112, -114, -116, -118, -120, -122, -124, -126, -128, -130, -132, -134, -136, -138, -140
Offset: 1
Keywords
Links
- Bela Bajnok, Additive Combinatorics: A Menu of Research Problems, arXiv:1705.07444 [math.NT], May 2017. See Table in Section 1.6.4.
Programs
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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, 3, 6], {n, 1, 70}]
Formula
Conjectures from Chai Wah Wu, Jun 10 2025: (Start)
a(n) = 2*a(n-1) - a(n-2) for n > 11.
G.f.: x*(-x^10 + x^9 - 2*x^8 - 2*x^7 + 4*x^6 - 2*x^5 - 2*x^4 + 4*x^3 - 2*x^2 + 17*x - 17)/(x - 1)^2. (End)