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A064180
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Composite numbers k such that the sum of the proper divisors of k not including 1, (Chowla's function, A048050) and their product (A007956) are both perfect squares.
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1
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117, 208, 292, 320, 475, 539, 549, 567, 873, 964, 1737, 2107, 2692, 2997, 3573, 3904, 4477, 4802, 5275, 5284, 5968, 6057, 7267, 7488, 7492, 9189, 9457, 9475, 10084, 10377, 11072, 11728, 11737, 12717, 13769, 14373, 14692, 16219, 16399, 17397
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OFFSET
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1,1
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LINKS
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EXAMPLE
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117 is in the sequence because the divisors of 117 are 1, 3, 9, 13, 39 and 117. Being squarefree itself, the product of divisors is a perfect square. The sum of the divisors in question, 3+9+13+39 = 64 and it is a perfect square.
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MATHEMATICA
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Select[ Range[2, 25000], IntegerQ[ Sqrt[ Apply[ Plus, Delete[ Divisors[ # ], -1]] - 1]] && IntegerQ[ Sqrt[ Apply[ Times, Delete[ Divisors[ # ], -1]]]] && ! PrimeQ[ # ] & ]
aQ[n_] := CompositeQ[n] && IntegerQ[Sqrt[n^(DivisorSigma[0, n]/2 - 1)]] && IntegerQ[Sqrt[DivisorSigma[1, n] - 1 - n]]; Select[Range[18000], aQ] (* Amiram Eldar, Jul 03 2019 *)
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PROG
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(Magma) [k:k in [1..18000]| not IsPrime(k) and IsSquare((&+Divisors(k))-1-k) and IsSquare((&*Divisors(k))/k) ]; // Marius A. Burtea, Jul 03 2019
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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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STATUS
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approved
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