A289290 -id:A289290 - cp's OEIS frontend

A289384 Numbers k such that the sum of the divisors of k is of the form m^3 + 1.

1, 12, 68, 82, 100, 730, 886, 1089, 1241, 1252, 1352, 1440, 1908, 2804, 2947, 3274, 5598, 6078, 7414, 9123, 10135, 10164, 10804, 10809, 11143, 12756, 13456, 13468, 15004, 21025, 23810, 24642, 25123, 26912, 26983, 34976, 37020, 40477, 45946, 48126, 55964, 56764

Offset: 1

Michel Lagneau, Jul 04 2017

nonn

Perfect squares in the sequence are 1, 100, 1089, 13456, 21025, ...

			730 is in the sequence because sigma(730) = 1332 = 11^3 + 1.

Robert G. Wilson v, Table of n, a(n) for n = 1..1510

Cf. A000203, A001093, A289290.

a:= proc(n) option remember; local k;
      for k from 1+`if`(n=1, 0, a(n-1)) while (t->t<>
      iroot(t, 3)^3)(numtheory[sigma](k)-1) do od; k
    end:
seq(a(n), n=1..40);  # Alois P. Heinz, Jul 04 2017

fQ[n_] := ! PrimeQ@n && Block[{sd = DivisorSigma[1, n]}, IntegerQ[(sd - 1)^(1/3)]]; Select[Range@59323, fQ] (* Robert G. Wilson v, Jul 05 2017 *)

isok(n) = ispower(sigma(n)-1, 3); \\ Michel Marcus, Jul 05 2017

More terms from Alois P. Heinz, Jul 04 2017

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.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Aug 20 2018

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

Cf. A001157, A046655, A289290.

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

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

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

cp's OEIS Frontend

A289384 Numbers k such that the sum of the divisors of k is of the form m^3 + 1.

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