A118372 S-perfect numbers.

6, 24, 28, 96, 126, 224, 384, 496, 1536, 1792, 6144, 8128, 14336, 15872, 24576, 98304, 114688, 393216, 507904, 917504, 1040384, 1572864, 5540590, 6291456, 7340032, 9078520, 16252928, 22528935, 25165824, 33550336, 56918394, 58720256, 100663296, 133169152

Offset: 1

Author

Vladeta Jovovic, May 15 2006

Keywords

• nonn

Comments

In base 12 the sequence becomes 6, 20, 24, 80, X6, 168, 280, 354, X80, 1054, 3680, 4854, 8368, 9228, 12280, 48X80, 56454, where X is 10 and E is 11. The perfect numbers (A000396) in this sequence in base 12 are 6, 24, 354, 4854. - Walter Kehowski, May 20 2006

Subsequence of A083207. - Reinhard Zumkeller, Oct 28 2010

Conjecture: If k is an S-perfect number, then A000203(k)/2 is a Zumkeller number (A083207). - Ivan N. Ianakiev, Apr 23 2017

Called "Granville numbers" by De Koninck (2009), after Andrew Granville, who proposed the problem of calculating these numbers in December 1996. - Amiram Eldar, Aug 11 2023

Examples

			2 is in S since s = Sum_{d|2, d<2 and d in S} d = 1 and 1 <= 2. Similarly, 3, 4, 5, 6 are in S with 6 as the first element such that s = n, that is, 6 is the first S-perfect number. - _Walter Kehowski_, May 20 2006
		

References

• Jean-Marie De Koninck, Those Fascinating Numbers, American Mathematical Society, 2009.

Links

• Donovan Johnson, Table of n, a(n) for n = 1..40 (terms < 4*10^9)
• Jean-Marie De Koninck and Aleksandar Ivić, On a sum of divisors problem, Publications de l'Institut Mathématique (Beograd), New Series, Vol. 64 (78) (1998), pp. 9-20.
• Gérard Villemin, Nombres S-PARFAITS ou Nombres de Granville, NOMBRES - Curiosités, théorie et usages, 2019 (in French).
• Wikipedia, Granville number.

Crossrefs

Subsequence of A023196 and A083207.

A000396 is a subsequence.

Cf. A000203, A181487.

Programs

C

#include  #include  #define MAX_SIZE_SSET 1000000 int main(int argc, char*argv[]) { int Sset[MAX_SIZE_SSET] ; int Ssetsize= 1; Sset[0]=1 ; for(int n=2; n < MAX_SIZE_SSET; n++) { int dsum=0 ; for(int i=0; i< Ssetsize; i++) { if( n % Sset[i] ==0 && Sset[i] < n) dsum += Sset[i] ; if (dsum > n || Sset[i] >=n) break ; } if( dsum <= n) { if(dsum==n) printf("%d\n",n) ; Sset[Ssetsize++ ]= n ; } } } /* R. J. Mathar, Oct 28 2010 */

Haskell

a118372_list = sPerfect 1 [] where
   sPerfect x ss | v > x = sPerfect (x + 1) ss
                 | v < x = sPerfect (x + 1) (x : ss)
                 | otherwise = x : sPerfect (x + 1) (x : ss)
                 where v = sum (filter ((== 0) . mod x) ss)
-- Reinhard Zumkeller, Oct 28 2010, Nov 02 2010, Feb 25 2012

Maple

with(numtheory); S:={1}: SP:=[]: for w to 1 do for n from 1 to 2*10^5 do d:=select(proc(z) z in S and zWalter Kehowski, May 20 2006

Mathematica

S = {1}; SP = {}; Do[ s = Total[ Intersection[S , Divisors[n]]]; If[s <= n, S = Union[S, {n}]]; If[s == n, Print[n]; AppendTo[SP, n]] , {n, 2, 2*10^5} ]; SP (* Jean-François Alcover, Dec 06 2011, after Walter Kehowski *)

Sage

def S_perfect_list(search_limit):
    S = []; T = []
    for n in (1..search_limit):
        d = [t for t in divisors(n) if t in S and t < n]
        s = sum(d)
        if s <= n: S.append(n)
        if s == n: T.append(n)
    return T
S_perfect_list(25555) # after Walter Kehowski, Peter Luschny, Sep 03 2018

Formula

S = {1}. Assume n>1 and that all numbers mWalter Kehowski, May 20 2006

I take the preceding comment to mean: S_0 = {1}. s_n = Sum_{d|n, d n, and S_{n-1} U {n} if s_n <= n. - Hugo van der Sanden, Oct 28 2010

Extensions

More terms from R. J. Mathar, May 17 2006, a(18) and a(19) Oct 28 2010

Two more terms added and C-program reduced by R. J. Mathar, Oct 28 2010

More terms from William Rex Marshall, Oct 28 2010