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

1,2

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.

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[#] &]

Crossrefs

Intersection of A005153 and A260653.

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

Cf. A325781, A325795.

Sequence in context: A365270 A238443 A325795 * A325781 A231565 A191743

Adjacent sequences: A359417 A359418 A359419 * A359421 A359422 A359423

Keyword

nonn

Author

Amiram Eldar, Dec 31 2022

Status

approved