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%I #11 Jan 08 2013 03:29:35
%S 2,4,8,27,22,340,44,156,62,1065,112,2467,158,1914,2551,4234,274,2161,
%T 344,8643,6611,12696,508,8410,522,28171,566,7500,814,39433,932,15000,
%U 57160,26980,15681,13590,1334,121327,7786,8908,1642,15896,1808,150069,74267,16105,2164
%N Consider the array T(n, m) where the n-th row is the sequence of integer coefficients of A(x), where 1<=a(n)<=n, such that A(x)^(1/n) consists entirely of integer coefficients and where m is the (m+1)-th coefficient. This is the row sum of A to the first coefficient of one.
%C mod(a(n),n):0,0,2,3,2,4,2,4,8,5,2,7,2,10,1,10,2,1,2,3,17,2,2,10,22,13,26,24,2, ...,.
%C The sum of just the even terms of T(n,k): 0,2,0,22,0,290,0,144,0,900,0,2288,0,1606,332,4124,0,1708,0,7908,790,10940,0,8196,0,24168,0,6920,0, ...,.
%F Sum_{m=0..k} from T(n, m), k is the least k>0 such that T(n, m)=1.
%F Sum_{m=0..A112283(n)} T(n, m).
%F a(p)=p(p-1)+2.
%t f[n_] := Module[{j = 1, a}, a[0] = 1; a[l_] := a[l] = Block[{k = 1, s = Sum[a[i]*x^i, {i, 0, l - 1}]}, While[ IntegerQ[ Last[ CoefficientList[ Series[(s + k*x^l)^(1/n), {x, 0, l}], x]]] != True, k++ ]; k]; While[a[j] != 1, j++ ]; Sum[ a[i], {i, 0, j}]]; Do[ Print[ f[n]], {n, 29}]
%Y Cf. A111603, A112283.
%K nonn
%O 1,1
%A _Paul D. Hanna_ and _Robert G. Wilson v_, Aug 29 2005
%E More terms from _Robert G. Wilson v_, Jul 25 2008