A112283 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.
1, 2, 3, 8, 5, 72, 7, 24, 9, 130, 11, 264, 13, 168, 210, 304, 17, 162, 19, 540, 378, 682, 23, 456, 25, 1274, 27, 336, 29, 1770, 31, 544, 2013, 918, 525, 504, 37, 3724, 234, 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
Offset: 1
Keywords
Examples
Only the pertinent part of the 'Table' in A109626.
{1, 1},
{1, 2, 1},
{1, 3, 3, 1},
{1, 4, 2, 4, 3, 4, 4, 4, 1},
{1, 5, 5, 5, 5, 1},
{1, 6, 3, 2, 3, 6, 6, 6, 3, 4, 6, 6, 6, 6, ..., },
{1, 7, 7, 7, 7, 7, 7, 1},
{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},
{1, 9, 9, 3, 9, 9, 3, 9, 9, 1},
{1, 10, 5, 10, 10, 2, 5, 10, 10, 10, 3, 10, 5, 10, ..., },
{1, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 1},
{1, 12, 6, 4, 9, 12, 4, 12, 12, 8, 6, 12, 6, 12, ..., },
{1, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 1}, ...,.
Programs
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Mathematica
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}]
Formula
Conjecture: a(n)=n if n is 1, a prime or the square of the odd primes.
Extensions
a(30)-a(50) from Robert G. Wilson v, Oct 29 2007
a(51)-a(65) from Robert G. Wilson v, Jul 25 2008
Comments