A346555 7-Sondow numbers: numbers k such that p^s divides k/p + 7 for every prime power divisor p^s of k.

1, 2, 6, 15, 78, 294, 12642, 539026980558, 7822623145454471412762

Offset

1,2

Comments

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

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

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

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

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

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

Contains 7*t for each primary pseudoperfect number t (A054377) that is divisible by 7. Terms of this form: a(6) = 7*A054377(3) and a(7) = 7*A054377(4). - Max Alekseyev, Nov 20 2025

Links

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

Github, Jonathan Sondow (1943 - 2020)

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

José María Grau Ribas, 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.

Mathematica

Sondow[mu_][n_]:=Sondow[mu][n]=Module[{fa=FactorInteger[n]}, IntegerQ[mu/n+Sum[1/fa[[i, 1]], {i, Length[fa]}]]]
Select[Range[10000000], Sondow[7][#]&]

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, this sequence, A346556, A346557.

Sequence in context: A307180 A009455 A244443 * A356223 A319336 A007709

Adjacent sequences: A346552 A346553 A346554 * A346556 A346557 A346558

Keyword

nonn,more

Author

José María Grau Ribas, Feb 04 2022

Extensions

a(8) from Martin Ehrenstein, Feb 04 2022

a(9) from Max Alekseyev, Nov 20 2025

Status

approved