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).
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.
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
KEYWORD
nonn
AUTHOR
José María Grau Ribas, Nov 10 2021
STATUS
approved