A346557 9-Sondow numbers: numbers k such that p^s divides k/p + 9 for every prime power divisor p^s of k.

1, 2, 5, 22, 54, 378, 16254, 423522, 19930521798, 472458569418

Offset: 1

Author

José María Grau Ribas, Feb 04 2022

Keywords

• nonn
• more

Comments

Numbers k such that A235137(k) == 9 (mod k).

A positive integer k is a 9-Sondow number if satisfies any of the following equivalent properties:

1) p^s divides k/p + 9 for every prime power divisor p^s of k.

2) 9/k + Sum_{prime p|k} 1/p is an integer.

3) 9 + Sum_{prime p|k} k/p == 0 (mod k).

4) Sum_{i=1..k} i^phi(k) == 9 (mod k).

Other numbers in the sequence: 19930521798, 472458569418, 76413794252037195696360941349774

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], 2013-2014.

Crossrefs

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

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

1-Sondow to 9-Sondow numbers: A349193, A330068, A346551, A346552, A346553, A346554, A346555, A346556, this sequence.

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[1000000],Sondow[9][#]&]

Extensions

a(9)-a(10) verified by Martin Ehrenstein, Feb 04 2022