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A346551 3-Sondow numbers: numbers k such that p^s divides k/p + 3 for every prime power divisor p^s of k.

%I A346551 #20 Jan 02 2022 10:02:59
%S A346551 1,2,10,18,126,5418,141174,6643507266,157486189806
%N A346551 3-Sondow numbers: numbers k such that p^s divides k/p + 3 for every prime power divisor p^s of k.
%C A346551 Numbers k such that A235137(k) == 3 (mod k).
%C A346551 A positive integer k is a 3-Sondow number if satisfies any of the following equivalent properties:
%C A346551 1) p^s divides k/p + 3 for every prime power divisor p^s of k.
%C A346551 2) 3/k + Sum_{prime p|k} 1/p is an integer.
%C A346551 3) 3 + Sum_{prime p|k} k/p == 0 (mod k).
%C A346551 4) Sum_{i=1..k} i^phi(k) == 3 (mod k).
%H A346551 Github, <a href="https://jonathansondow.github.io/">Jonathan Sondow (1943 - 2020)</a>
%H A346551 J. M. Grau, A. M. Oller-Marcén, and D. Sadornil, <a href="https://arxiv.org/abs/2111.14211">On µ-Sondow Numbers</a>, arXiv:2111.14211 [math.NT], 2021.
%H A346551 J. M. Grau, A. M. Oller-Marcen  and J. Sondow, <a href="https://arxiv.org/abs/1309.7941">On the congruence 1^n + 2^n +... + n^n = d (mod n), where d divides n</a>, arXiv:1309.7941 [math.NT], 2013-2014.
%t A346551 Sondow[mu_][n_]:= Sondow[mu][n]= Module[{fa=FactorInteger[n]},IntegerQ[mu/n+Sum[1/fa[[i,1]],{i,Length[fa]}]]]
%t A346551 Select[Range[1000000],Sondow[3][#]&]
%Y A346551 Cf. A054377, A007850, A235137, A348058, A348059.
%Y A346551 (-1) and (-2) -Sondow numbers: A326715, A330069.
%Y A346551 1-Sondow to 9-Sondow numbers: A349193, A330068, A346551, A346552, A346553, A346554, A346555, A346556, A346557.
%K A346551 nonn,more
%O A346551 1,2
%A A346551 _José María Grau Ribas_, Dec 04 2021
%E A346551 a(8)-a(9) from _Martin Ehrenstein_, Dec 31 2021