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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, 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 #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 (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. Cf. A000203, A181487. Sequence in context: A364977 A336641 A336550 * A263928 A219362 A226476 Adjacent sequences: A118369 A118370 A118371 * A118373 A118374 A118375 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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