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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 sequence is the first occurrence of 1 after the first column.
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%I #12 Mar 15 2015 20:23:33

%S 1,2,3,8,5,72,7,24,9,130,11,264,13,168,210,304,17,162,19,540,378,682,

%T 23,456,25,1274,27,336,29,1770,31,544,2013,918,525,504,37,3724,234,

%U 280,41,504,43,4180,2025,414,47,816,49,1300,3723,1196,53,972,550,1960,3933,986,59,6480,61,1798,4095,1792,3055

%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 sequence is the first occurrence of 1 after the first column.

%C The degree >0 of the polynomial above whose coefficient is 1.

%F Conjecture: a(n)=n if n is 1, a prime or the square of the odd primes.

%e Only the pertinent part of the 'Table' in A109626.

%e {1, 1},

%e {1, 2, 1},

%e {1, 3, 3, 1},

%e {1, 4, 2, 4, 3, 4, 4, 4, 1},

%e {1, 5, 5, 5, 5, 1},

%e {1, 6, 3, 2, 3, 6, 6, 6, 3, 4, 6, 6, 6, 6, ..., },

%e {1, 7, 7, 7, 7, 7, 7, 1},

%e {1, 8, 4, 8, 2, 8, 4, 8, 7, 8, 8, 8, 4, 8, 8, 8, 3, 8, 8, 8, 2, 8, 8, 8, 1},

%e {1, 9, 9, 3, 9, 9, 3, 9, 9, 1},

%e {1, 10, 5, 10, 10, 2, 5, 10, 10, 10, 3, 10, 5, 10, ..., },

%e {1, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 1},

%e {1, 12, 6, 4, 9, 12, 4, 12, 12, 8, 6, 12, 6, 12, ..., },

%e {1, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 1}, ...,.

%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++ ]; j]; Table[ f[n], {n, 10}]

%Y Cf. A109626, A112284.

%K nonn

%O 1,2

%A _Paul D. Hanna_ and _Robert G. Wilson v_, Aug 20 2005

%E a(30)-a(50) from _Robert G. Wilson v_, Oct 29 2007

%E a(51)-a(65) from _Robert G. Wilson v_, Jul 25 2008