cp's OEIS Frontend

A220019 Number of cyclotomic cosets of 7 mod 10^n.

%I A220019 #27 Jul 03 2023 23:21:59
%S A220019 4,27,93,265,685,1265,2005,2905,3965,5185,6565,8105,9805,11665,13685,
%T A220019 15865,18205,20705,23365,26185,29165,32305,35605,39065,42685,46465,
%U A220019 50405,54505,58765,63185,67765,72505,77405,82465,87685,93065,98605,104305,110165,116185,122365,128705,135205,141865,148685
%N A220019 Number of cyclotomic cosets of 7 mod 10^n.
%H A220019 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3, -3, 1).
%F A220019 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]
%F A220019 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]
%e A220019 a(2) = 27 because there are 27 cyclotomic cosets of 7 mod 100:
%e A220019 {1, 7, 49, 43}
%e A220019 {2, 14, 98, 86}
%e A220019 {3, 21, 47, 29}
%e A220019 {4, 28, 96, 72}
%e A220019 {5, 35, 45, 15}
%e A220019 {6, 42, 94, 58}
%e A220019 {8, 56, 92, 44}
%e A220019 {9, 63, 41, 87}
%e A220019 {10, 70, 90, 30}
%e A220019 {11, 77, 39, 73}
%e A220019 {12, 84, 88, 16}
%e A220019 {13, 91, 37, 59}
%e A220019 {17, 19, 33, 31}
%e A220019 {18, 26, 82, 74}
%e A220019 {20, 40, 80, 60}
%e A220019 {22, 54, 78, 46}
%e A220019 {23, 61, 27, 89}
%e A220019 {24, 68, 76, 32}
%e A220019 {25, 75}
%e A220019 {34, 38, 66, 62}
%e A220019 {36, 52, 64, 48}
%e A220019 {50}
%e A220019 {51, 57, 99, 93}
%e A220019 {53, 71, 97, 79}
%e A220019 {55, 85, 95, 65}
%e A220019 {67, 69, 83, 81}
%e A220019 {0}
%t A220019 a[n_] := DivisorSum[10^n, EulerPhi[#] / MultiplicativeOrder[7, #] & ]; Array[a, 50] (* _Jean-François Alcover_, Dec 18 2015 *)
%o A220019 (PARI) for(n=1,50,print1(sumdiv(10^n, d, eulerphi(d)/znorder(Mod(7, d)))", "))
%Y A220019 Cf. A006694, A220468.
%K A220019 base,nonn
%O A220019 1,1
%A A220019 _V. Raman_, Jan 27 2013