A346553 5-Sondow numbers: numbers k such that p^s divides k/p + 5 for every prime power divisor p^s of k.

1, 2, 3, 14, 66, 1974, 307146, 3270666, 42404405538, 318501038226

Offset: 1

Author

José María Grau Ribas, Jan 16 2022

Keywords

• nonn
• more

Comments

Numbers k such that A235137(k) == 5 (mod k).

A positive integer k is a 5-Sondow number if satisfies any of the following equivalent properties:

1) p^s divides k/p + 5 for every prime power divisor p^s of k.

2) 5/k + Sum_{prime p|k} 1/p is an integer.

3) 5 + Sum_{prime p|k} k/p == 0 (mod k).

4) Sum_{i=1..k} i^phi(k) == 5 (mod k).

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, A346557.

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[10^7], Sondow[5][#]&]

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 + 5) % p^j, return(0)));); return(1);} \\ Michel Marcus, Jan 17 2022

Extensions

a(9)-a(10) from Martin Ehrenstein, Jan 19 2022