OFFSET
1,12
COMMENTS
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Graph of terms k in row n of A381801 not in row n of A381799, n = 1..60.
FORMULA
a(p^m) = 0 for prime p and m >= 0.
a(n) >= 1 for n in A024619, since residue 0 (mod n) is in T(n) is not in any V(n,p) and thus also not in S(n), because n is not a prime power.
EXAMPLE
n a(n) T(n) \ S(n)
----------------------------------------------
6 1 {0}
10 1 {0}
12 2 {0,6}
18 3 {0,6,12}
20 2 {0,10}
24 4 {0,6,12,18}
28 2 {0,14}
30 8 {0,6,10,12,15,18,20,24}
36 5 {0,6,12,18,24}
72 8 {0,6,12,18,24,36,48,54}
100 7 {0,10,20,40,50,60,80}
108 12 {0,6,12,18,24,36,48,54,60,72,84,96}
144 11 {0,6,12,18,24,36,48,54,72,96,108}
210 70 {0,6,10,12,14,15,18,20,..,200,204}
.
a(2) = 0 since T(2) = S(2) = V(2,2) = {0,1}.
a(4) = 0 since T(4) = S(4) = V(4,2) = {0,1,2}.
a(6) = 1 since T(6) = {0,1,2,3,4} but S(6) = {1,2,4} U {1,3}.
a(12) = 2 since T(12) = {0,1,2,3,4,6,8,9} but S(12) = {1,2,4,8} U {1,3,9}.
a(16) = 0 since T(16) = S(16) = V(16,2) = {0,1,2,4,8}.
a(18) = 3 since T(18) = {0,1,2,3,4,6,8,9,10,12,14,16} but S(18) = {1,2,4,8,16,14,10} U {1,3,9}. The numbers {0,6,12} do not appear in S(18).
a(30) = 8 since T(30) = {0,1,2,3,4,5,6,8,9,10,12,15,16,18,20,21,24,25,27}, but S(30) = {1,2,4,8,16} U {1,3,9,27,21} U {1,5,25}. The numbers {0,6,12,18,24} do not appear in S(30), etc.
MATHEMATICA
f[x_, p_] := Block[{m = 2, r, c},
Which[PrimePowerQ[x],
Join[{0}, #1^Range[0, #2 - 1]] & @@ FactorInteger[x][[1]],
PowerMod[p, m, x] == p, {1, p}, True, c[_] := False;
c[1] = c[p] = True; {1, p}~Join~
Reap[While[r = PowerMod[p, m, x]; ! c[r], Sow[r];
c[r] = True; m++]][[-1, 1]]]];
g[x_] := Block[{c, ff, m, r, p, s, w},
c[_] := True; ff = FactorInteger[x][[All, 1]]; w = Length[ff]; s = {1};
Do[Set[p[i], ff[[i]]], {i, w}];
Do[Set[s, Union@ Flatten@ Join[s, #[[-1, 1]]]] &@ Reap@
Do[m = s[[j]];
While[Sow@ Set[r, Mod[m*p[i], x]];
c[r], c[r] = False; m *= p[i]],
{j, Length[s]}], {i, w}];
Length[s] ];
{0}~Join~Table[g[n] - CountDistinct@ Flatten@ Map[f[n, #] &, FactorInteger[n][[All, 1]] ], {n, 2, 120}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael De Vlieger, Apr 12 2025
STATUS
approved