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

1, 2, 6, 42, 1806, 47058, 2214502422, 52495396602, 8490421583559688410706771261086

Offset: 1

Author

José María Grau Ribas, Nov 10 2021

Keywords

• nonn

Comments

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

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.

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

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

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

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

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

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

Links

• Github, Jonathan Sondow (1943 - 2020)
• J. M. Grau, A. M. Oller-Marcén, and D. Sadornil, On µ-Sondow Numbers, arXiv:2111.14211 [math.NT], 2021.
• J. M. Grau, A. M. Oller-Marcen and J. Sondow, On the congruence 1^n + 2^n + ... + n^n = d (mod n), where d divides n, arXiv:1309.7941 [math.NT], 2013.

Crossrefs

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

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

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

Programs

Mathematica

Sondow[mu_][n_]:= Sondow[mu][n]= Module[{fa=FactorInteger[n]},IntegerQ[mu/n+Sum[1/fa[[i,1]],{i,Length[fa]}]]];
Select[Range[100000],Sondow[1][#]&]