A220020 Number of cyclotomic cosets of 9 mod 10^n.

6, 20, 56, 140, 256, 404, 584, 796, 1040, 1316, 1624, 1964, 2336, 2740, 3176, 3644, 4144, 4676, 5240, 5836, 6464, 7124, 7816, 8540, 9296, 10084, 10904, 11756, 12640, 13556, 14504, 15484, 16496, 17540, 18616, 19724, 20864, 22036, 23240, 24476, 25744, 27044, 28376, 29740, 31136, 32564, 34024, 35516, 37040, 38596

Offset: 1

Author

V. Raman, Jan 27 2013

Keywords

• base
• nonn

Examples

			a(2) = 20 because there are 20 cyclotomic cosets of 9 mod 100:
{1, 9, 81, 29, 61, 49, 41, 69, 21, 89}
{3, 27, 43, 87, 83, 47, 23, 7, 63, 67}
{11, 99, 91, 19, 71, 39, 51, 59, 31, 79}
{13, 17, 53, 77, 93, 37, 33, 97, 73, 57}
{2, 18, 62, 58, 22, 98, 82, 38, 42, 78}
{4, 36, 24, 16, 44, 96, 64, 76, 84, 56}
{6, 54, 86, 74, 66, 94, 46, 14, 26, 34}
{8, 72, 48, 32, 88, 92, 28, 52, 68, 12}
{10, 90}
{30, 70}
{20, 80}
{40, 60}
{50}
{5, 45}
{15, 35}
{55, 95}
{65, 85}
{25}
{75}
{0}
		

Links

• Index entries for linear recurrences with constant coefficients, signature (3, -3, 1).

Crossrefs

Cf. A006694, A220468.

Programs

Mathematica

a[n_] := DivisorSum[10^n, EulerPhi[#]/MultiplicativeOrder[9, #]&]; Array[a, 50] (* Jean-François Alcover, Dec 10 2015, adapted from PARI *)
LinearRecurrence[{3,-3,1},{6,20,56,140,256},50] (* Harvey P. Dale, Jul 12 2025 *)

PARI

for(n=1,50,print1(sumdiv(10^n, d, eulerphi(d)/znorder(Mod(9, d)))", "))

Formula

Conjecture: a(n) = 4*(4*n^2-7*n-1) for n>2. a(n) = 3*a(n-1)-3*a(n-2)+a(n-3) for n>5. G.f.: 2*x*(8*x^4-13*x^3-7*x^2-x-3) / (x-1)^3. - Colin Barker, Apr 13 2013