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A084377
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a(n) = n^3 + 7.
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3
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7, 8, 15, 34, 71, 132, 223, 350, 519, 736, 1007, 1338, 1735, 2204, 2751, 3382, 4103, 4920, 5839, 6866, 8007, 9268, 10655, 12174, 13831, 15632, 17583, 19690, 21959, 24396, 27007, 29798, 32775, 35944, 39311, 42882, 46663, 50660, 54879, 59326
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OFFSET
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0,1
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COMMENTS
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These numbers cannot be perfect squares. - Cino Hilliard, Sep 03 2006
Proof that n^3+7 <> k^2 for all integers n,k.
Assume y^2 - x^3 = 7 has an integer solution.
Modulo 4, we have {0,1,0,1} - {0,1,0,3} == 3; y is even and x is odd.
y^2+1 = x^3+8 = (x+2) [(x-1)^2+3]. Let z = (x-1)^2+3 == 3 mod 4.
The 1-mod-4 numbers are closed under multiplication, so z has a prime factor p == 3 mod 4.
That p divides y^2+1; y^2 == -1 mod p.
But (quadratic reciprocity) there is no square root of -1 modulo p.
That refutes the assumption.
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LINKS
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FORMULA
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a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>3. - Vincenzo Librandi, Jun 10 2016
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MATHEMATICA
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PROG
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(PARI) a(n) = n^3 + 7;
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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