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, 5998279018951962402

Offset

1,2

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).

8490421583559688410706771261086 (previously in data) and 35979351189199316534587473905773572006 (discovered by Han M. Wang) are also terms. - José María Grau Ribas, Jun 03 2026

Links

Table of n, a(n) for n=1..9.

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-Marcén and Jonathan Sondow, On the congruence 1^m + 2^m + ... + m^m == n (mod m) with n | m, arXiv:1309.7941 [math.NT], 2013-2014.

Han Wang, Port Fillings for Primary Pseudoperfect Numbers, arXiv:2605.21518 [math.NT], 2026.

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][#]&]

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.

Sequence in context: A123137 A014117 A054377 * A230311 A379686 A276416

Adjacent sequences: A349190 A349191 A349192 * A349194 A349195 A349196

Keyword

nonn,hard,more

Author

José María Grau Ribas, Nov 10 2021

Extensions

a(9) discovered by Han M. Wang and corrected by José María Grau Ribas, May 22 2026

Status

approved