A336503 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].

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, 63, 64, 70, 72, 80, 84, 90, 96, 100, 105, 108, 112, 120, 124, 126, 128, 132, 135, 136, 140, 144, 147, 150, 154, 156, 160, 162, 165, 168, 176, 180, 182, 186, 189, 192

Offset

1,2

Comments

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.

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).

The number of terms not exceeding 10^k for k = 1, 2, ... are 6, 34, 243, 1790, 14703, 120276, 1030279, ...

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Paul Pollack and Lola Thompson, On the degrees of divisors of T^n-1>, New York Journal of Mathematics, Vo. 19 (2013), pp. 91-116, preprint, arXiv:1206.2084 [math.NT], 2012.

Lola Thompson, Products of distinct cyclotomic polynomials, Ph.D. thesis, Dartmouth College, 2012.

Lola Thompson, On the divisors of x^n - 1 in F_p[x], International Journal of Number Theory, Vol. 9, No. 2 (2013), pp. 421-430.

Lola Thompson, Variations on a question concerning the degrees of divisors of x^n - 1, Journal de Théorie des Nombres de Bordeaux, Vol. 26, No. 1 (2014), pp. 253-267.

Eric Weisstein's World of Mathematics, Finite Field.

Wikipedia, Finite field.

Mathematica

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] &]

Crossrefs

Cf. A000265, A002326, A007733, A007814.

Cf. A000010, A260653, A336504, A336505.

Sequence in context: A129121 A018556 A018387 * A275717 A029449 A028815

Adjacent sequences: A336500 A336501 A336502 * A336504 A336505 A336506

Keyword

nonn

Author

Amiram Eldar, Jul 23 2020

Status

approved