A330069 (-2)-Sondow numbers: numbers m such that p divides m/p - 2 for every prime divisor p of m.

1, 4, 60, 1716, 3444, 132396, 4428816612, 48846257124, 865498410347676, 29474266940021148, 1101686782618260636, 488394001964999430175732692

Offset

1,2

Comments

Also, numbers k such that Sum_{i=1..k} i^A000010(k) == -2 (mod k).

Contains as a subsequence 2 * even terms of A007850 (Giuga numbers). All currently listed terms > 4 are of this form. Further terms include 1108159829234141602577157118356 and 3821334362841015969111519832677012.

Links

Table of n, a(n) for n=1..12.

Mathematica

G[n_, k_] := G[n, k] = Mod[Sum[PowerMod[i, k, n], {i, 1, n}], n];
Select[Range[2000], G[#, EulerPhi[#]] == n-2 &]
fa=FactorInteger;
se[n_, k_] := Select[Transpose[fa[n]][[1]], IntegerQ[k/(# - 1)] &];
sumlis[li_] := Sum[li[[i]], {i, 1, Length[li]}]
Table[If[Mod[-n/se[n, EulerPhi[n]] // sumlis, n] == n-2, n], {n, 1,
   1000000}] // Union

Prog

(PARI) isok(n) = sumdiv(n, d, eulerphi(n/d) * Mod(d, n)^eulerphi(n)) == -2; \\ Daniel Suteu, Jan 13 2020

Crossrefs

k-Sondow numbers: A326715 (k=-1), A330068 (k=2), A346551 (k=3), A346552 (k=4), A346553 (k=5), A346554 (k=6), A346555 (k=7), A346556 (k=8), A346557 (k=9).

Cf. A000010, A054377, A007850.

Sequence in context: A098630 A377629 A336637 * A211309 A013502 A303286

Adjacent sequences: A330066 A330067 A330068 * A330070 A330071 A330072

Keyword

nonn,hard,more

Author

José María Grau Ribas, Nov 30 2019

Extensions

a(7)-a(8) from Giovanni Resta, Feb 27 2020

Edited and a(9)-a(12) added by Max Alekseyev, Nov 10 2025

Status

approved