login
Number of cyclotomic cosets of 11 mod 10^n.
0

%I #28 Jul 03 2023 23:20:00

%S 10,27,65,119,189,275,377,495,629,779,945,1127,1325,1539,1769,2015,

%T 2277,2555,2849,3159,3485,3827,4185,4559,4949,5355,5777,6215,6669,

%U 7139,7625,8127,8645,9179,9729,10295,10877,11475,12089,12719,13365,14027,14705,15399,16109,16835,17577,18335,19109,19899

%N Number of cyclotomic cosets of 11 mod 10^n.

%C How is this related to A181890? - _R. J. Mathar_, Apr 11 2013

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3, -3, 1).

%F Conjecture: a(n) = 8*n^2-2*n-1 for n>1. a(n) = 3*a(n-1)-3*a(n-2)+a(n-3) for n>4. G.f.: x*(5*x^3-14*x^2+3*x-10) / (x-1)^3. - _Colin Barker_, Apr 13 2013

%e a(2) = 27 because there are 27 cyclotomic cosets of 11 mod 100:

%e {1, 11, 21, 31, 41, 51, 61, 71, 81, 91}

%e {3, 33, 63, 93, 23, 53, 83, 13, 43, 73}

%e {7, 77, 47, 17, 87, 57, 27, 97, 67, 37}

%e {9, 99, 89, 79, 69, 59, 49, 39, 29, 19}

%e {2, 22, 42, 62, 82}

%e {12, 32, 52, 72, 92}

%e {4, 44, 84, 24, 64}

%e {14, 54, 94, 34, 74}

%e {6, 66, 26, 86, 46}

%e {16, 76, 36, 96, 56}

%e {8, 88, 68, 48, 28}

%e {18, 98, 78, 58, 38}

%e {5, 55}

%e {15, 65}

%e {25, 75}

%e {35, 85}

%e {45, 95}

%e {0}

%e {10}

%e {20}

%e {30}

%e {40}

%e {50}

%e {60}

%e {70}

%e {80}

%e {90}

%t a[n_] := DivisorSum[10^n, EulerPhi[#] / MultiplicativeOrder[11, #] &]; Array[a, 50] (* _Jean-François Alcover_, Dec 18 2015 *)

%o (PARI) for(n=1,50,print1(sumdiv(10^n, d, eulerphi(d)/znorder(Mod(11, d)))", "))

%Y Cf. A006694, A220468.

%K base,nonn

%O 1,1

%A _V. Raman_, Jan 27 2013