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A118372 S-perfect numbers. 14
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
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
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.
FORMULA
S = {1}. Assume n>1 and that all numbers m<n have already been tested. Let s = Sum_{d|n, d<n and d in S} d. If s<=n, then n is now in S. The paper linked to above has some characterization results. - Walter Kehowski, May 20 2006
I take the preceding comment to mean: S_0 = {1}. s_n = Sum_{d|n, d<n and d in S_{n-1}} d. Then S_n := S_{n-1} if s_n > n, and S_{n-1} U {n} if s_n <= n. - Hugo van der Sanden, Oct 28 2010
EXAMPLE
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
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 z<n end, divisors(n)); s:=convert(d, `+`); if s<=n then S:=S union {n} fi; if s=n then SP:=[op(SP), n]; print(n); fi; od; od; SP; # Walter 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 *)
PROG
(C) #include <stdlib.h> #include <stdio.h> #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
(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
CROSSREFS
Subsequence of A023196 and A083207.
A000396 is a subsequence.
Sequence in context: A364977 A336641 A336550 * A263928 A219362 A226476
KEYWORD
nonn,changed
AUTHOR
Vladeta Jovovic, May 15 2006
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
STATUS
approved

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Last modified June 13 23:50 EDT 2024. Contains 373391 sequences. (Running on oeis4.)