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

Author

Amiram Eldar, Jul 23 2020

Keywords

• nonn

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.

Crossrefs

Subsequence of A013929.

Complement of A260653 with respect to A336506.

Programs

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 *)