A359420 Numbers that are both practical (A005153) and phi-practical (A260653).

1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 30, 32, 36, 40, 42, 48, 54, 56, 60, 64, 72, 80, 84, 90, 96, 100, 108, 112, 120, 126, 128, 132, 140, 144, 150, 156, 160, 162, 168, 176, 180, 192, 198, 200, 208, 210, 216, 220, 224, 234, 240, 252, 256, 260, 264, 270, 272, 280, 288

Offset: 1

Author

Amiram Eldar, Dec 31 2022

Keywords

• nonn

Comments

First differs from A325795 at n = 45, and from A325781 at n = 36.

Numbers k such that each number in the range 1..sigma(k) is a sum of distinct divisors of k, and each number in the range 1..k is a subsum of the multiset {phi(d) : d | k}.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Andreas Weingartner, Uniform distribution of alpha*n modulo one for a family of integer sequences, Uniform Distribution Theory, Vol. 18, No. 2 (2023), pp. 19-30; arXiv preprint, arXiv:2303.16819 [math.NT], 2023. Mentions this sequence.

Crossrefs

Intersection of A005153 and A260653.

Cf. A000010 (phi), A000203 (sigma).

Cf. A325781, A325795.

Programs

Mathematica

f[p_, e_] := (p^(e + 1) - 1)/(p - 1); pracQ[n_] := (ind = Position[(fct = FactorInteger[n])[[;; , 1]]/(1 + FoldList[Times, 1, f @@@ Most@fct]), _?(# > 1 &)]) == {};
phiPracticalQ[n_] := 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)]; (* Frank M Jackson's code at A260653 *)
Select[Range[300], pracQ[#] && phiPracticalQ[#] &]