Numbers n such that the polynomial x^n - 1 has a divisor of every degree up to n.

1, 2, 3, 4, 6, 8, 12, 15, 16, 18, 20, 24, 30, 32, 36, 40, 42, 48, 54, 56, 60, 64, 72, 80, 84, 90, 96, 100, 105, 108, 112, 120, 126, 128, 132, 140, 144, 150, 156, 160, 162, 165, 168, 176, 180, 192, 195, 198, 200, 208, 210, 216, 220, 224, 234, 240, 252, 255

1

Computing terms beyond 1872 is difficult using the algorithm below because the number of subsets becomes very large. This sequence does not appear in the Pomerance et al paper.

T. D. Noe, Plot of 284 terms

T. D. Noe, Table of 284 terms

Carl Pomerance, Lola Thompson, and Andreas Weingartner, On integers n for which X^n−1 has a divisor of every degree, arXiv 1511.03357 (Nov 11 2015)

(Mma) t = {}; Do[pwrs = Length /@ CoefficientList[Transpose[Rest[FactorList[x^pwr - 1]]][[1]], x] - 1; If[Max[Rest[pwrs]/Most[pwrs]] <= 2 || Complement[Range[pwr], Union[Total /@ Subsets[pwrs]]] == {}, AppendTo[t, pwr]], {pwr, 500}]; t

nonn,hard,nice

T. D. Noe, Nov 12 2015