A336506 Lambda-practical numbers: numbers that are p-practical for every rational prime p.

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, 84, 90, 96, 100, 105, 108, 112, 120, 126, 128, 132, 135, 140, 144, 150, 156, 160, 162, 165, 168, 176, 180, 192, 195, 198, 200, 208, 210, 216, 220, 224, 225, 234, 240

Offset: 1

Author

Amiram Eldar, Jul 23 2020

Keywords

• nonn

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. See A336503, A336504 and A336505 for examples.

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).

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).

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.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Lola Thompson, Products of distinct cyclotomic polynomials, Ph.D. thesis, Dartmouth College, 2012.
• 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.

Crossrefs

Disjoint union of A260653 and A336507.

Cf. A336503, A336504, A336505.

Cf. A000010, A002322, A005117.

Programs

Mathematica

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]