A336505 - OEIS

%I #9 Jul 26 2020 02:34:25

%S 1,2,3,4,5,6,8,10,12,15,16,18,20,24,25,30,32,35,36,39,40,42,44,45,48,

%T 50,52,54,55,56,60,62,64,65,66,70,72,75,78,80,84,88,90,93,96,100,104,

%U 105,108,110,112,117,120,124,125,126,128,130,132,135,140,144,150

%N 5-practical numbers: numbers m such that the polynomial x^m - 1 has a divisor of every degree <= m in the prime field F_5[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 5-practical if and only if every number 1 <= k <= m can be written as Sum_{d|m} A007736(d) * n_d, where A007736(d) is the multiplicative order of 5 modulo the largest divisor of d not divisible by 5, and 0 <= n_d <= phi(d)/A007736(d).

%C The number of terms not exceeding 10^k for k = 1, 2, ... are 7, 46, 286, 2179, 16847, 141446, 1223577, ...

%H Amiram Eldar, <a href="/A336505/b336505.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[#, 5] &]

%Y Cf. A132739, A007736, A112765.

%Y Cf. A000010, A260653, A336503, A336504.

%K nonn

%O 1,2

%A _Amiram Eldar_, Jul 23 2020