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A279550
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Composite numbers m for which L(p(i)/p(j)) = -1 for all i, j, where p(k) are the prime factors of m and L(x/y) is the Legendre symbol of x and y, defined to be 1 if x is a quadratic residue (mod y) and -1 if x is a quadratic non-residue (mod y).
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1
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6, 10, 12, 15, 18, 20, 22, 24, 26, 30, 35, 36, 38, 40, 44, 45, 48, 50, 51, 52, 54, 58, 60, 65, 72, 74, 75, 76, 80, 85, 86, 87, 88, 90, 91, 96, 100, 104, 106, 108, 115, 116, 118, 119, 120, 122, 123, 130, 134, 135, 143, 144, 148, 150, 152, 153, 159, 160, 162
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OFFSET
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1,1
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COMMENTS
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L(x/y) = L(y/x) for primes x, y, and either x=4r+1 for some r, or y=4r+1 for some r.
We observe pairs of the form (a(n), a(n)+1) with a(n): 35, 44, 50, 51, 74, 75, 85, 86, 87, 90, 115, 118, 119, 122, 134, 143, ...
We observe triples of the form (a(n), a(n)+1, a(n)+2) with a(n): 50, 74, 85, 86, 118, 174, 214, 260, 286, 318, 324, 403, 484, 634, 635, ...
We observe quadruplets of the form (a(n), a(n)+1, a(n)+2, a(n)+3) with a(n): 85, 634, 635, 696, 842, 1340, 1382, 2424, 2599, 2929, 3145, 3576, ...
We observe quintuplets of the form (a(n), a(n)+1, a(n)+2, a(n)+3, a(n)+4) with a(n): 634, 10951, 12184, 20281, 21967, 29493, ...
We observe sextuplets of the form (a(n), a(n)+1, a(n)+2, a(n)+3, a(n)+4), a(n)+5) with a(n): 60747, 742582, ...
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LINKS
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EXAMPLE
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10 is in the sequence because the prime factors are 2 and 5 => L(2,5) = L(5,2) = -1.
275070 is in the sequence because the prime factors are 2, 3, 5, 53 and 173 => L(2,3) = L(2,5) = L(2,53) = L(2,173) = L(3,5) = L(3,53) = L(3,173) = L(5,53) = L(5,173) = L(53,173) = L(173,53) = L(173,5) = L(173,3) = L(173,2) = L(53,5) = L(53,3) = L(53,2) = L(5,3) = L(5,2) = L(3,2) = -1.
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MATHEMATICA
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fQ[n_] := If[!PrimeQ@ n, Block[{pf = Transpose[ FactorInteger[n]][[1]]}, lng = Length@ pf; Union[ Flatten[ Table[ JacobiSymbol[pf[[i]], pf[[j]]], {i, lng}, {j, lng}]]] == {-1, 0}], False]; Select[ Range@ 330, fQ]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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