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A069098
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Number of minimal monic annihilator polynomials over the ring of integers modulo n.
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1
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1, 1, 4, 1, 2, 1, 4, 27, 8, 1, 12, 1, 32, 9, 256, 1, 432, 1, 16, 81, 512, 1, 12, 3125, 2048, 19683, 256, 1, 72, 1, 65536, 6561, 32768, 25, 1728, 1, 131072, 59049, 32, 1, 2592, 1, 65536, 135, 2097152, 1, 6912, 823543, 800000
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OFFSET
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2,3
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LINKS
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Table of n, a(n) for n=2..50.
A. Vardy, Comments and C program
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FORMULA
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a(n) = 1 if n is prime.
Let n = prod_{i=1}^m p_i^e_i be the prime decomposition of n. For p prime and integers k, q, define N(k, p, q) = p^{ sum_{j=0}^{k-1} b(j)} where b(j) is the largest integer b in {0, 1, 2, ..., q } such that p^b divides j!. Then a(n) = prod_{i=1}^m N(S(n), p_i, e_i) where S(n) is the n-th Smarandache number (sequence A002034), i.e. S(n) is the smallest integer k such that n divides k!. - Navin Kashyap (nkashyap(AT)ece.ucsd.edu), Aug 07 2002
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EXAMPLE
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a(6)=2 because there are exactly two minimal annihilator polynomials over Z_6, namely X^3 + 5x and X^3 + 3x^2 + 2x.
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MATHEMATICA
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A002034[n_] := Module[ {m = 1}, While[ !IntegerQ[m!/n], m++]; m]; b[j_, p_, q_] := Module[ {m = q}, While[ !Divisible[j!, p^m], m--]; m]; nn[k_, p_, q_] := p^Sum[b[j, p, q], {j, 0, k-1}]; a[n_?PrimeQ] = 1; a[n_] := Module[ {fi, m, pp, ee}, fi = FactorInteger[n]; m = Length[fi]; pp = fi[[All, 1]]; ee = fi[[All, 2]]; Product[ nn[ A002034[n], pp[[i]], ee[[i]]], {i, 1, m}]]; Table[ a[n], {n, 2, 50}] (* From Jean-François Alcover, Dec 19 2011, after Navin Kashyap *)
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CROSSREFS
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Cf. A002034.
Sequence in context: A109008 A187025 A074695 * A126241 A019777 A090885
Adjacent sequences: A069095 A069096 A069097 * A069099 A069100 A069101
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KEYWORD
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nonn,nice
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AUTHOR
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Alexander Vardy (vardy(AT)montblanc.ucsd.edu), Apr 05 2002
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EXTENSIONS
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More terms from Navin Kashyap (nkashyap(AT)ece.ucsd.edu), Aug 07 2002
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STATUS
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approved
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