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A349193 1-Sondow numbers: numbers j such that p divides j/p + 1 for every prime divisor p of j.

%I A349193 #48 Dec 29 2021 07:10:22
%S A349193 1,2,6,42,1806,47058,2214502422,52495396602,
%T A349193 8490421583559688410706771261086
%N A349193 1-Sondow numbers: numbers j such that p divides j/p + 1 for every prime divisor p of j.
%C A349193 These are the weak primary pseudoperfect numbers mentioned in Grau-Oller-Sondow (2013).
%C A349193 Includes the primary pseudoperfect numbers (A054377). Any weak primary pseudoperfect number which is not a primary pseudoperfect number must have more than 58 distinct prime factors, and therefore must be greater than 10^110; none are known.
%C A349193 A positive integer j is a k-Sondow number if satisfies any of the following equivalent properties:
%C A349193 1) p^s divides j/p + k for every prime power divisor p^s of j.
%C A349193 2) k/j + Sum_{prime p|j} 1/p is an integer.
%C A349193 3) k + Sum_{prime p|j} j/p == 0 (mod j).
%C A349193 4) Sum_{i=1..j} i^A000010(j) == k (mod j).
%C A349193 Numbers m such that A235137(m) == 1 (mod m).
%H A349193 Github, <a href="https://jonathansondow.github.io/"> Jonathan Sondow (1943 - 2020)</a>
%H A349193 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 A349193 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.
%t A349193 Sondow[mu_][n_]:= Sondow[mu][n]= Module[{fa=FactorInteger[n]},IntegerQ[mu/n+Sum[1/fa[[i,1]],{i,Length[fa]}]]];
%t A349193 Select[Range[100000],Sondow[1][#]&]
%Y A349193 Cf. A000010, A054377, A007850, A235137, A348058, A348059.
%Y A349193 (-1) and (-2)-Sondow numbers: A326715, A330069.
%Y A349193 2-Sondow to 9-Sondow numbers: A330068, A346551, A346552, A346553, A346554, A346555, A346556, A346557.
%K A349193 nonn
%O A349193 1,2
%A A349193 _José María Grau Ribas_, Nov 10 2021