A220019 Number of cyclotomic cosets of 7 mod 10^n.
4, 27, 93, 265, 685, 1265, 2005, 2905, 3965, 5185, 6565, 8105, 9805, 11665, 13685, 15865, 18205, 20705, 23365, 26185, 29165, 32305, 35605, 39065, 42685, 46465, 50405, 54505, 58765, 63185, 67765, 72505, 77405, 82465, 87685, 93065, 98605, 104305, 110165, 116185, 122365, 128705, 135205, 141865, 148685
Offset: 1
Examples
a(2) = 27 because there are 27 cyclotomic cosets of 7 mod 100:
{1, 7, 49, 43}
{2, 14, 98, 86}
{3, 21, 47, 29}
{4, 28, 96, 72}
{5, 35, 45, 15}
{6, 42, 94, 58}
{8, 56, 92, 44}
{9, 63, 41, 87}
{10, 70, 90, 30}
{11, 77, 39, 73}
{12, 84, 88, 16}
{13, 91, 37, 59}
{17, 19, 33, 31}
{18, 26, 82, 74}
{20, 40, 80, 60}
{22, 54, 78, 46}
{23, 61, 27, 89}
{24, 68, 76, 32}
{25, 75}
{34, 38, 66, 62}
{36, 52, 64, 48}
{50}
{51, 57, 99, 93}
{53, 71, 97, 79}
{55, 85, 95, 65}
{67, 69, 83, 81}
{0}
Links
- Index entries for linear recurrences with constant coefficients, signature (3, -3, 1).
Programs
-
Mathematica
a[n_] := DivisorSum[10^n, EulerPhi[#] / MultiplicativeOrder[7, #] & ]; Array[a, 50] (* Jean-François Alcover, Dec 18 2015 *)
-
PARI
for(n=1,50,print1(sumdiv(10^n, d, eulerphi(d)/znorder(Mod(7, d)))", "))
Formula
Empirical G.f.: x*(88*x^5-142*x^4-63*x^3-24*x^2-15*x-4) / (x-1)^3. [Colin Barker, Feb 03 2013]
Conjecture: a(n) = 5*(16*n^2-60*n+37) for n>3. a(n) = 3*a(n-1)-3*a(n-2)+a(n-3) for n>6. [Colin Barker, Apr 14 2013]