A346554 6-Sondow numbers: numbers k such that p^s divides k/p + 6 for every prime power divisor p^s of k.

1, 4, 7, 9, 20, 36, 252, 10836, 282348, 13287014532, 314972379612

Offset

1,2

Comments

Numbers k such that A235137(k) == 6 (mod k).

A positive integer k is a 6-Sondow number if satisfies any of the following equivalent properties:

1) p^s divides k/p + 6 for every prime power divisor p^s of k.

2) 6/k + Sum_{prime p|k} 1/p is an integer.

3) 6 + Sum_{prime p|k} k/p == 0 (mod k).

4) Sum_{i=1..k} i^phi(k) == 6 (mod k).

Other numbers in the sequence: 13287014532, 314972379612, 50942529501358130464240627566516

Links

Table of n, a(n) for n=1..11.

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.

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[6][#]&]

Prog

(PARI) isok(k) = {my(f=factor(k)); for (i=1, #f~, my(p=f[i, 1]); for (j=1, f[i, 2], if ((k/p + 6) % p^j, return(0))); ); return(1); } \\ Michel Marcus, Jan 17 2022

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, A346557.

Sequence in context: A281144 A103073 A166742 * A022541 A330695 A063798

Adjacent sequences: A346551 A346552 A346553 * A346555 A346556 A346557

Keyword

nonn,more

Author

José María Grau Ribas, Jan 16 2022

Extensions

a(10)-a(11) verified by Martin Ehrenstein, Jan 21 2022

Status

approved