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%I #10 Jul 26 2020 02:34:18
%S 1,2,3,4,6,8,12,14,15,16,18,20,21,24,28,30,32,36,40,42,45,48,54,56,60,
%T 63,64,70,72,80,84,90,96,100,105,108,112,120,124,126,128,132,135,136,
%U 140,144,147,150,154,156,160,162,165,168,176,180,182,186,189,192
%N 2-practical numbers: numbers m such that the polynomial x^m - 1 has a divisor of every degree <= m in the prime field F_2[x].
%C For a rational prime number p, a "p-practical number" is a number m such that the polynomial x^m - 1 has a divisor of every degree <= m in F_p[x], the prime field of order p.
%C A number m is 2-practical if and only if every number 1 <= k <= m can be written as Sum_{d|m} A007733(d) * n_d, where A007733(d) is the multiplicative order of 2 modulo the odd part of d, and 0 <= n_d <= phi(d)/A007733(d).
%C The number of terms not exceeding 10^k for k = 1, 2, ... are 6, 34, 243, 1790, 14703, 120276, 1030279, ...
%H Amiram Eldar, <a href="/A336503/b336503.txt">Table of n, a(n) for n = 1..10000</a>
%H Paul Pollack and Lola Thompson, <a href="http://nyjm.albany.edu/j/2013/19-7.html">On the degrees of divisors of T^n-1></a>, New York Journal of Mathematics, Vo. 19 (2013), pp. 91-116, <a href="https://arxiv.org/abs/1206.2084">preprint</a>, arXiv:1206.2084 [math.NT], 2012.
%H Lola Thompson, <a href="http://www.lolathompson.com/uploads/1/1/0/6/110629329/thesis.pdf">Products of distinct cyclotomic polynomials</a>, Ph.D. thesis, Dartmouth College, 2012.
%H Lola Thompson, <a href="https://doi.org/10.1142/S1793042112501412">On the divisors of x^n - 1 in F_p[x]</a>, International Journal of Number Theory, Vol. 9, No. 2 (2013), pp. 421-430.
%H Lola Thompson, <a href="https://doi.org/10.5802/jtnb.866">Variations on a question concerning the degrees of divisors of x^n - 1</a>, Journal de Théorie des Nombres de Bordeaux, Vol. 26, No. 1 (2014), pp. 253-267.
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FiniteField.html">Finite Field</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Finite_field">Finite field</a>.
%t rep[v_, c_] := Flatten @ Table[ConstantArray[v[[i]], {c[[i]]}], {i, Length[c]}]; mo[n_, p_] := MultiplicativeOrder[p, n/p^IntegerExponent[n, p]]; ppQ[n_, p_] := Module[{d = Divisors[n]}, m = mo[#, p] & /@ d; ns = EulerPhi[d]/m; r = rep[m, ns]; Min @ Rest @ CoefficientList[Series[Product[1 + x^r[[i]], {i, Length[r]}], {x, 0, n}], x] > 0]; Select[Range[200], ppQ[#, 2] &]
%Y Cf. A000265, A002326, A007733, A007814.
%Y Cf. A000010, A260653, A336504, A336505.
%K nonn
%O 1,2
%A _Amiram Eldar_, Jul 23 2020
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