A336506 - OEIS

%I #12 Aug 04 2020 03:56:22

%S 1,2,3,4,6,8,12,15,16,18,20,24,30,32,36,40,42,45,48,54,56,60,64,72,80,

%T 84,90,96,100,105,108,112,120,126,128,132,135,140,144,150,156,160,162,

%U 165,168,176,180,192,195,198,200,208,210,216,220,224,225,234,240

%N Lambda-practical numbers: numbers that are p-practical for every rational prime p.

%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. See A336503, A336504 and A336505 for examples.

%C A number m is lambda-practical if and only if every number 1 <= k <= m can be written as Sum_{d|m} lambda(d) * n_d, where lambda(d) = A002322(d) is the Carmichael lambda function, and 0 <= n_d <= phi(d)/lambda(d).

%C A squarefree number is lambda-practical if and only if it is phi-practical (A260653). All phi-practical numbers are lambda-practical, but there are infinitely many lambda-practical numbers that are not phi-practical: 45, 135, 225, 405, 675, ... (A336507).

%C If N(x) is the number of terms not exceeding, x then there are two positive constants c_1 and c_2 such that c_1 * x/log(x) <= N(x) <= c_2 * x/log(x) for all x >= 2.

%H Amiram Eldar, <a href="/A336506/b336506.txt">Table of n, a(n) for n = 1..10000</a>

%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/10.5802/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.

%t rep[v_, c_] := Flatten @ Table[ConstantArray[v[[i]], {c[[i]]}], {i, Length[c]}]; lpQ[n_] := Module[{d = Divisors[n], lam, ns, r, x}, lam = CarmichaelLambda[d]; ns = EulerPhi[d]/lam; r = rep[lam, ns]; Min @ Rest @ CoefficientList[Series[Product[1 + x^r[[i]], {i, Length[r]}], {x, 0, n}], x] > 0]; Select[Range[250], lpQ]

%Y Disjoint union of A260653 and A336507.

%Y Cf. A336503, A336504, A336505.

%Y Cf. A000010, A002322, A005117.

%K nonn

%O 1,2

%A _Amiram Eldar_, Jul 23 2020