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. 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.
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]
CROSSREFS
KEYWORD
nonn
AUTHOR
Amiram Eldar, Jul 23 2020
STATUS
approved