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A136585
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Solutions of an a*x+b*y=c Prime Diophantine Equation: Prime[m]x+Prime[m+1]*y=Prime[m-1] : as Abs[n*Prime[m]] Or Abs[n*Prime[m+1]] in x+y*n=Prime[m-1].
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0
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2, 4, 5, 6, 9, 20, 33, 35, 42, 44, 57, 68, 104, 114, 117, 119, 145, 174, 279, 301, 310, 322, 345, 376, 410, 430, 517, 533, 590, 649, 740, 777, 976, 1159, 1537, 1590, 2345, 2412
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OFFSET
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1,1
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COMMENTS
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Starting at the second prime 3, solutions are obtained to the Equation
x+y*n=Prime[m-1]
or
n=(Prime[m-1]+x)/y
Either n*Prime[m]/or n*Prime[m+1] is an Integer.
using the Wagon Diophantine solver Module for n and then the specific prime that it is a rational number of is multiplied out to give an Integer.
The resulting numbers are made positive and sorted for magnitude
to give the output sequence.
This sequence is an effort to get some sequence related by the primes:
{Prime[m-1],Prime[m],Prime[m+1]}
by
Prime[m]x+Prime[m+1]*y=Prime[m-1]
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REFERENCES
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A Course in Computational Number Theory by Bressoud and Wagon,2001
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LINKS
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Table of n, a(n) for n=1..38.
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FORMULA
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a[out]=Abs[If[ IntegerQ[n*Prime[m+1]],n*Prime[m+1] else n*Prime[m]]] where n is a rational number: n=(Prime[m-1]+x)/y Sequence is sorted by magnitude.
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MATHEMATICA
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Clear[n, m, l] DiophantineSolve[{a_, b_}, c_, n_] := Module[{d, e}, {d, e} = ExtendedGCD[a, b]; If[Mod[c, d] == 0, Transpose[{c*e, {b, -a}}/d].{1, n}, {}]]; a = Table[Table[Simplify[If[l == 2, Prime[m], Prime[m + 1]]*(n /. Solve[DiophantineSolve[{Prime[m], Prime[m + 1]}, Prime[m - 1], n][[l]] - Prime[m - 1] == 0, n])], {l, 2, 1, -1}], {m, 2, 20}]; Union[Abs[Flatten[a]]]
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CROSSREFS
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Sequence in context: A056635 A163116 A003306 * A122721 A014224 A175342
Adjacent sequences: A136582 A136583 A136584 * A136586 A136587 A136588
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KEYWORD
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nonn,uned
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AUTHOR
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Roger L. Bagula, Mar 26 2008
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STATUS
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approved
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