1,2

Numbers k such that A235137(k) == 8 (mod k).

A positive integer k is a 8-Sondow number if satisfies any of the following equivalent properties:

1) p^s divides k/p + 8 for every prime power divisor p^s of k.

2) 8/k + Sum_{prime p|k} 1/p is an integer.

3) 8 + Sum_{prime p|k} k/p == 0 (mod k).

4) Sum_{i=1..k} i^phi(k) == 8 (mod k).

a(10) <= 8 * A054377(8) = 67923372668477507285654170088688.

a(10) > 10^25. - Max Alekseyev, Dec 04 2025

Github, Jonathan Sondow (1943 - 2020)

José María Grau, A. M. Oller-Marcén and D. Sadornil, On µ-Sondow Numbers, arXiv:2111.14211 [math.NT], 2021.

José María Grau, A. M. Oller-Marcen and Jonathan Sondow, On the congruence 1^n + 2^n + ... + n^n = d (mod n), where d divides n, arXiv:1309.7941 [math.NT], 2013-2014.

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

Select[Range[400000], Sondow[8][#]&]

Contains 8 * even terms of A054377 as subsequence.

Cf. A007850, A235137, A348058, A348059.

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

1-Sondow to 9-Sondow numbers: A349193, A330068, A346551, A346552, A346553, A346554, A346555, this sequence, A346557.

Sequence in context: A362007 A212564 A222843 * A004320 A389329 A389377

Adjacent sequences: A346553 A346554 A346555 * A346557 A346558 A346559

nonn,more

José María Grau Ribas, Feb 04 2022

a(8)-a(9) verified by Martin Ehrenstein, Feb 04 2022

approved