A346555 - OEIS

%I #10 Mar 18 2022 00:28:26

%S 1,2,6,15,78,294,12642,539026980558

%N 7-Sondow numbers: numbers k such that p^s divides k/p + 7 for every prime power divisor p^s of k.

%C Numbers k such that A235137(k) == 7 (mod k).

%C A positive integer k is a 7-Sondow number if satisfies any of the following equivalent properties:

%C 1) p^s divides k/p + 7 for every prime power divisor p^s of k.

%C 2) 7/k + Sum_{prime p|k} 1/p is an integer.

%C 3) 7 + Sum_{prime p|k} k/p == 0 (mod k).

%C 4) Sum_{i=1..k} i^phi(k) == 7 (mod k).

%H Github, <a href="https://jonathansondow.github.io/">Jonathan Sondow (1943 - 2020)</a>

%H J. M. Grau, A. M. Oller-Marcén and D. Sadornil, <a href="https://arxiv.org/abs/2111.14211">On µ-Sondow Numbers</a>, arXiv:2111.14211 [math.NT], 2021.

%H J. M. Grau, A. M. Oller-Marcen and J. Sondow, <a href="https://arxiv.org/abs/1309.7941">On the congruence 1^n + 2^n + ... + n^n = d (mod n), where d divides n</a>, arXiv:1309.7941 [math.NT], 2013-2014.

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

%t Select[Range[10000000],Sondow[7][#]&]

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

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

%Y 1-Sondow to 9-Sondow numbers: A349193, A330068, A346551, A346552, A346553, A346554, this sequence, A346556, A346557.

%K nonn,more

%O 1,2

%A _José María Grau Ribas_, Feb 04 2022

%E a(8) from _Martin Ehrenstein_, Feb 04 2022