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A346557 9-Sondow numbers: numbers k such that p^s divides k/p + 9 for every prime power divisor p^s of k.

%I A346557 #8 Mar 18 2022 00:28:43
%S A346557 1,2,5,22,54,378,16254,423522,19930521798,472458569418
%N A346557 9-Sondow numbers: numbers k such that p^s divides k/p + 9 for every prime power divisor p^s of k.
%C A346557 Numbers k such that A235137(k) == 9 (mod k).
%C A346557 A positive integer k is a 9-Sondow number if satisfies any of the following equivalent properties:
%C A346557 1) p^s divides k/p + 9 for every prime power divisor p^s of k.
%C A346557 2) 9/k + Sum_{prime p|k} 1/p is an integer.
%C A346557 3) 9 + Sum_{prime p|k} k/p == 0 (mod k).
%C A346557 4) Sum_{i=1..k} i^phi(k) == 9 (mod k).
%C A346557 Other numbers in the sequence: 19930521798, 472458569418, 76413794252037195696360941349774
%H A346557 Github, <a href="https://jonathansondow.github.io/">Jonathan Sondow (1943 - 2020)</a>
%H A346557 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 A346557 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 A346557 Sondow[mu_][n_]:=Sondow[mu][n]=Module[{fa=FactorInteger[n]},IntegerQ[mu/n+Sum[1/fa[[i,1]],{i,Length[fa]}]]]
%t A346557 Select[Range[1000000],Sondow[9][#]&]
%Y A346557 Cf. A054377, A007850, A235137, A348058, A348059.
%Y A346557 (-1) and (-2) -Sondow numbers: A326715, A330069.
%Y A346557 1-Sondow to 9-Sondow numbers: A349193, A330068, A346551, A346552, A346553, A346554, A346555, A346556, this sequence.
%K A346557 nonn,more
%O A346557 1,2
%A A346557 _José María Grau Ribas_, Feb 04 2022
%E A346557 a(9)-a(10) verified by _Martin Ehrenstein_, Feb 04 2022