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.

6, 40, 136, 2696, 3352, 46976, 223736, 5509736, 1915798072

Offset: 1

Author

Amiram Eldar, Aug 20 2018

Keywords

• nonn
• more

Comments

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

Crossrefs

Cf. A001157, A046655, A289290.

Programs

Magma

[n: n in [2..6*10^6] |not IsPrime(n) and IsSquare(DivisorSigma(2, n)-1)]; // Vincenzo Librandi, Aug 22 2018

Mathematica

sQ[n_] := IntegerQ[Sqrt[n]]; aQ[n_] := CompositeQ[n] && sQ[DivisorSigma[2,n]-1]; Select[Range[10000],aQ]

PARI

forcomposite(n=2, 1e15, if( issquare(sigma(n,2)-1), print1(n, ", ")))