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A176996 Numbers n such that sum of divisors, sigma(n), and sum of the proper divisors, sigma(n)-n, are both square. 1
1, 3, 119, 527, 935, 3591, 3692, 6887, 12319, 47959, 65151, 97767, 99116, 202895, 237900, 373319, 438311, 699407, 734111, 851927, 957551, 1032156, 1064124, 1437599, 1443959, 2858687, 3509231, 3699311, 4984199, 7237415 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

The only prime in this sequence is 3. All prime numbers have the square 1 as the sum of their proper divisors. But since 3 is the only prime of the form n^2 - 1, it is the only prime that satisfies the first condition for inclusion in this sequence.

LINKS

Donovan Johnson, Table of n, a(n) for n = 1..300

Antonio Roldan Martinez, La suma de sus divisores es cuadrado perfecto

FORMULA

Intersection of A006532 and A073040.

EXAMPLE

119 has divisors 1, 7, 17, 119; it is in the list because 1+7+17+119 = 144 = 12^2 and 1+7+17 = 25 = 5^2.

MATHEMATICA

Intersection[Select[Range[10^5], IntegerQ[Sqrt[-# + Plus@@Divisors[#]]] &], Select[Range[10^5], IntegerQ[Sqrt[Plus@@Divisors[#]]] &]] (* Alonso del Arte, Dec 08 2010 *)

t = {}; Do[If[And @@ IntegerQ /@ Sqrt[{x = DivisorSigma[1, n], x - n}], AppendTo[t, n]], {n, 10^6}]; t (* Jayanta Basu, Jul 27 2013 *)

sdQ[n_]:=Module[{d=DivisorSigma[1, n]}, AllTrue[{Sqrt[d], Sqrt[d-n]}, IntegerQ]]; Select[Range[73*10^5], sdQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Apr 17 2018 *)

PROG

(Sage) is_A176996 = lambda n: is_square(sigma(n)) and is_square(sigma(n)-n) # D. S. McNeil, Dec 09 2010

CROSSREFS

Cf. A006532, which considers all divisors; A048699, which for nonprime numbers considers all divisors except the number itself; A073040, which is the union of A048699 and the prime numbers (A000040).

Sequence in context: A172013 A143781 A114077 * A159522 A103743 A304016

Adjacent sequences:  A176993 A176994 A176995 * A176997 A176998 A176999

KEYWORD

nonn

AUTHOR

Claudio Meller, Dec 08 2010

STATUS

approved

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Last modified December 15 20:00 EST 2019. Contains 330000 sequences. (Running on oeis4.)