A382138 a(n) = A381800(n) - A381798(n).

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 1, 0, 0, 3, 0, 2, 1, 1, 0, 4, 0, 1, 0, 2, 0, 8, 0, 0, 1, 1, 1, 5, 0, 1, 1, 3, 0, 10, 0, 2, 3, 1, 0, 6, 0, 5, 1, 2, 0, 9, 1, 4, 1, 1, 0, 16, 0, 1, 2, 0, 1, 14, 0, 2, 1, 12, 0, 8, 0, 1, 5, 2, 1, 16, 0, 5, 0, 1, 0, 19, 1

Offset: 1

Author

Michael De Vlieger, Apr 12 2025

Keywords

• nonn

Comments

Number of residue classes r (mod n) of k such that rad(k) | n that are not residue classes q (mod n) of p^m, p | n.

Let S(n) = row n of A381799 and let T(n) = row n of A381801. Let V(n,p) = {p^m mod n : m >= 0}. Then S(n) = U_{p|n} V(n,p).

Examples

			    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.
		

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.

Crossrefs

Cf. A000961, A024619, A381798, A381799, A381800, A381801.

Programs

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}]

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.