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1-Sondow numbers: numbers j such that p divides j/p + 1 for every prime divisor p of j.
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%I #48 Dec 29 2021 07:10:22

%S 1,2,6,42,1806,47058,2214502422,52495396602,

%T 8490421583559688410706771261086

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

%C These are the weak primary pseudoperfect numbers mentioned in Grau-Oller-Sondow (2013).

%C 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 A positive integer j is a k-Sondow number if satisfies any of the following equivalent properties:

%C 1) p^s divides j/p + k for every prime power divisor p^s of j.

%C 2) k/j + Sum_{prime p|j} 1/p is an integer.

%C 3) k + Sum_{prime p|j} j/p == 0 (mod j).

%C 4) Sum_{i=1..j} i^A000010(j) == k (mod j).

%C Numbers m such that A235137(m) == 1 (mod m).

%H Github, <a href="https://jonathansondow.github.io/"> Jonathan Sondow (1943 - 2020)</a>

%H 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 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 Sondow[mu_][n_]:= Sondow[mu][n]= Module[{fa=FactorInteger[n]},IntegerQ[mu/n+Sum[1/fa[[i,1]],{i,Length[fa]}]]];

%t Select[Range[100000],Sondow[1][#]&]

%Y Cf. A000010, A054377, A007850, A235137, A348058, A348059.

%Y (-1) and (-2)-Sondow numbers: A326715, A330069.

%Y 2-Sondow to 9-Sondow numbers: A330068, A346551, A346552, A346553, A346554, A346555, A346556, A346557.

%K nonn

%O 1,2

%A _José María Grau Ribas_, Nov 10 2021