OFFSET
1,1
COMMENTS
Thompson (2012) proved that all phi-practical numbers are lambda-practical, that all the terms of this sequence are not squarefree numbers, and that this sequence is infinite: for example, 45 * Product_{i=10..k} prime(i) is a term for all k >= 10.
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..300
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
phiPracticalQ[n_] := If[n<1, False, If[n==1, True, (lst = Sort @ EulerPhi @ Divisors[n]; ok=True; Do[If[lst[[m]]>Sum[lst[[l]], {l, 1, m-1}]+1, (ok=False; Break[])], {m, 1, Length[lst]}]; ok)]]; rep[v_, c_] := Flatten @ Table[ConstantArray[v[[i]], {c[[i]]}], {i, Length[c]}]; lambdaPracticalQ[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[1000], !phiPracticalQ[#] && lambdaPracticalQ[#] &] (* after Frank M Jackson at A260653 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Amiram Eldar, Jul 23 2020
STATUS
approved