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A346554 6-Sondow numbers: numbers k such that p^s divides k/p + 6 for every prime power divisor p^s of k.

%I A346554 #16 Jan 21 2022 07:51:11
%S A346554 1,4,7,9,20,36,252,10836,282348,13287014532,314972379612
%N A346554 6-Sondow numbers: numbers k such that p^s divides k/p + 6 for every prime power divisor p^s of k.
%C A346554 Numbers k such that A235137(k) == 6 (mod k).
%C A346554 A positive integer k is a 6-Sondow number if satisfies any of the following equivalent properties:
%C A346554 1) p^s divides k/p + 6 for every prime power divisor p^s of k.
%C A346554 2) 6/k + Sum_{prime p|k} 1/p is an integer.
%C A346554 3) 6 + Sum_{prime p|k} k/p == 0 (mod k).
%C A346554 4) Sum_{i=1..k} i^phi(k) == 6 (mod k).
%C A346554 Other numbers in the sequence: 13287014532, 314972379612, 50942529501358130464240627566516
%H A346554 Github, <a href="https://jonathansondow.github.io/">Jonathan Sondow (1943 - 2020)</a>
%H A346554 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 A346554 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 A346554 Sondow[mu_][n_]:=Sondow[mu][n]=Module[{fa=FactorInteger[n]},IntegerQ[mu/n+Sum[1/fa[[i,1]],{i,Length[fa]}]]]
%t A346554 Select[Range[10000000],Sondow[6][#]&]
%o A346554 (PARI) isok(k) = {my(f=factor(k)); for (i=1, #f~, my(p=f[i,1]); for (j=1, f[i,2], if ((k/p + 6) % p^j, return(0)));); return(1);} \\ _Michel Marcus_, Jan 17 2022
%Y A346554 Cf. A054377, A007850, A235137, A348058, A348059.
%Y A346554 (-1) and (-2) -Sondow numbers: A326715, A330069.
%Y A346554 1-Sondow to 9-Sondow numbers: A349193, A330068, A346551, A346552, A346553, A346554, A346555, A346556, A346557.
%K A346554 nonn,more
%O A346554 1,2
%A A346554 _José María Grau Ribas_, Jan 16 2022
%E A346554 a(10)-a(11) verified by _Martin Ehrenstein_, Jan 21 2022