A346552 4-Sondow numbers: numbers k such that p^s divides k/p + 4 for every prime power divisor p^s of k.

1, 5, 8, 24, 168, 7224, 188232, 8858009688, 209981586408

Offset: 1

Author

José María Grau Ribas, Jan 11 2022

Keywords

• nonn
• more

Comments

Numbers k such that A235137(k) == 4 (mod k).

A positive integer k is a 4-Sondow number if satisfies any of the following equivalent properties:

1) p^s divides k/p + 4 for every prime power divisor p^s of k.

2) 4/k + Sum_{prime p|k} 1/p is an integer.

3) 4 + Sum_{prime p|k} k/p == 0 (mod k).

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

Other numbers in the sequence: 8858009688, 209981586408, 33961686334238753642827085044344

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[10000000],Sondow[4][#]&]

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 + 4) % p^j, return(0)));); return(1);} \\ Michel Marcus, Jan 17 2022

Extensions

a(8)-a(9) verified by Martin Ehrenstein, Jan 21 2022