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A318169
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Composite numbers k such that sigma_2(k) - 1 is a square, where sigma_2(k) = A001157(k) is the sum of squares of divisors of k.
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0
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OFFSET
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1,1
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COMMENTS
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This property is shared with all the primes since sigma_2(p) = 1 + p^2.
The values of sqrt(sigma_2(a(n)-1)) are 7, 47, 157, 3107, 3863, 54243, 257843, 6349657, 2207848187.
Are there terms not of the form 2^k * p where p is prime? - David A. Corneth, Aug 20 2018
2*10^12 < a(10) <= 44463118771144. The terms 21687324345660824, 14524130539077100050485512, 287674439504279743204606472 (and others) of the form 2^k * p can be found by solving the quadratic Diophantine equation sigma_2(2^k) * (p^2 + 1) = x^2 + 1 for appropriate values of k. - Giovanni Resta, Aug 20 2018
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LINKS
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MATHEMATICA
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sQ[n_] := IntegerQ[Sqrt[n]]; aQ[n_] := CompositeQ[n] && sQ[DivisorSigma[2, n]-1]; Select[Range[10000], aQ]
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PROG
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(PARI) forcomposite(n=2, 1e15, if( issquare(sigma(n, 2)-1), print1(n, ", ")))
(Magma) [n: n in [2..6*10^6] |not IsPrime(n) and IsSquare(DivisorSigma(2, n)-1)]; // Vincenzo Librandi, Aug 22 2018
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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