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

%I A359420 #18 Feb 04 2024 03:27:22
%S A359420 1,2,4,6,8,12,16,18,20,24,30,32,36,40,42,48,54,56,60,64,72,80,84,90,
%T A359420 96,100,108,112,120,126,128,132,140,144,150,156,160,162,168,176,180,
%U A359420 192,198,200,208,210,216,220,224,234,240,252,256,260,264,270,272,280,288
%N A359420 Numbers that are both practical (A005153) and phi-practical (A260653).
%C A359420 First differs from A325795 at n = 45, and from A325781 at n = 36.
%C A359420 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}.
%H A359420 Amiram Eldar, <a href="/A359420/b359420.txt">Table of n, a(n) for n = 1..10000</a>
%H A359420 Andreas Weingartner, <a href="https://pcwww.liv.ac.uk/~karpenk/JournalUDT/vol18/no2/02Weingart_pdf.pdf">Uniform distribution of alpha*n modulo one for a family of integer sequences</a>, Uniform Distribution Theory, Vol. 18, No. 2 (2023), pp. 19-30; <a href="https://arxiv.org/abs/2303.16819">arXiv preprint</a>, arXiv:2303.16819 [math.NT], 2023. Mentions this sequence.
%t A359420 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 &)]) == {};
%t A359420 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 *)
%t A359420 Select[Range[300], pracQ[#] && phiPracticalQ[#] &]
%Y A359420 Intersection of A005153 and A260653.
%Y A359420 Cf. A000010 (phi), A000203 (sigma).
%Y A359420 Cf. A325781, A325795.
%K A359420 nonn
%O A359420 1,2
%A A359420 _Amiram Eldar_, Dec 31 2022