A336507 Lambda-practical numbers (A336506) that are not phi-practical (A260653).

45, 135, 225, 405, 675, 765, 855, 1035, 1125, 1215, 1275, 1305, 1395, 1665, 1845, 1935, 2025, 2115, 2295, 2565, 3105, 3375, 3645, 3825, 3915, 4185, 4275, 4995, 5175, 5535, 5625, 5805, 6075, 6345, 6375, 6525, 6885, 6975, 7155, 7695, 7965, 8235, 8325, 9045, 9225

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

Subsequence of A013929.

Complement of A260653 with respect to A336506.

Sequence in context: A098924 A195318 A131011 * A156370 A044377 A044758

Adjacent sequences: A336504 A336505 A336506 * A336508 A336509 A336510

Keyword

nonn

Author

Amiram Eldar, Jul 23 2020

Status

approved