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%I A318169 #28 Aug 07 2024 13:01:42
%S A318169 6,40,136,2696,3352,46976,223736,5509736,1915798072
%N A318169 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.
%C A318169 This property is shared with all the primes since sigma_2(p) = 1 + p^2.
%C A318169 The values of sqrt(sigma_2(a(n))-1) are 7, 47, 157, 3107, 3863, 54243, 257843, 6349657, 2207848187.
%C A318169 Are there terms not of the form 2^k * p where p is prime? - _David A. Corneth_, Aug 20 2018
%C A318169 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
%t A318169 sQ[n_] := IntegerQ[Sqrt[n]]; aQ[n_] := CompositeQ[n] && sQ[DivisorSigma[2,n]-1]; Select[Range[10000],aQ]
%o A318169 (PARI) forcomposite(n=2, 1e15, if( issquare(sigma(n,2)-1), print1(n, ", ")))
%o A318169 (Magma) [n: n in [2..6*10^6] |not IsPrime(n) and IsSquare(DivisorSigma(2, n)-1)]; // _Vincenzo Librandi_, Aug 22 2018
%Y A318169 Cf. A001157, A046655, A289290.
%K A318169 nonn,more
%O A318169 1,1
%A A318169 _Amiram Eldar_, Aug 20 2018

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