A181487 -id:A181487 - cp's OEIS frontend

A364861 Numbers k such that k and k+1 are both S-abundant numbers (A181487).

5984, 7424, 21944, 39375, 56924, 77175, 82004, 84524, 89775, 109395, 116655, 158235, 174824, 180495, 185535, 188055, 193544, 200024, 209055, 235935, 238095, 240344, 245024, 250964, 256095, 261260, 262184, 263024, 266475, 279279, 282975, 283815, 294975, 297296

Offset: 1

Amiram Eldar, Aug 11 2023

nonn

De Koninck and Ivić found that the least number k such that k, k+1, and k+2 are 3 consecutive integers that are S-abundant numbers is 171078830 (which is also the first term of A096536).

Amiram Eldar, Table of n, a(n) for n = 1..10000
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.
Wikipedia, Granville number.

Subsequence of A181487.

Cf. A096399, A096536, A118372.

seq[kmax_] := Module[{s = {1}, a = {}, sum, q1 = False, q2}, Do[sum = Total[Select[Divisors[k], MemberQ[s, #] &]]; q2 = sum > k; If[!q2, AppendTo[s, k]]; If[q1 && q2, AppendTo[a, k-1]]; q1 = q2, {k, 2, kmax}]; a]; seq[40000]

lista(nmax) = {my(c = 0, s, q1 = 0, q2); for(n=2, nmax, if(sumdiv(n, d, !bittest(c, d)*d) > 2*n, c+=1<M. F. Hasler at A181487

A364862 S-weird numbers: S-abundant numbers (A181487) k such that no subset of the aliquot divisors of k that are in the set S sums to k, where S is the set defined in A118372.

Original entry on oeis.org

70, 836, 2704, 2744, 4030, 5530, 5810, 5830, 6230, 6790, 7070, 7192, 7210, 7490, 7630, 7910, 7912, 8890, 9170, 9272, 9590, 9730, 10430, 10570, 10792, 10990, 11410, 11690, 12110, 12530, 12670, 13370, 13510, 13790, 13930, 14770, 15610, 15890, 16030, 16310, 16730

Offset: 1

Amiram Eldar, Aug 11 2023

nonn

Analogous to weird numbers (A006037) as S-perfect numbers (A118372) are analogous to perfect numbers (A000396) and S-abundant numbers (A181487) are analogous to abundant numbers (A005101).

Apparently, includes all the weird numbers (verified for all terms below 5*10^7). It also includes additional terms: 2704, 2744, 5530, 5810, 6230, 6790, 7070, 7210, 7490, ... .

Amiram Eldar, Table of n, a(n) for n = 1..10000

Subsequence of A181487.

Cf. A000396, A005101, A006037, A118372.

weirdQ[n_, d_] := If[Total[d] <= n, False, SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, n}], n] == 0];
S = {1}; Sweird = {}; Do[s = Total[(d = Intersection[S, Divisors[n]])]; If[s <= n, AppendTo[S, n], If[weirdQ[n, d], AppendTo[Sweird, n]]], {n, 2, 10^4}]; Sweird

A118372 S-perfect numbers.

Original entry on oeis.org

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

Vladeta Jovovic, May 15 2006

nonn

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

			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

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

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.

Subsequence of A023196 and A083207.
A000396 is a subsequence.
Cf. A000203, A181487.

#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 */

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

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

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 *)

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

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

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

A364858 a(n) = Sum_{d|n, d < n, d in S} d, where S is the set defined in A118372.

Original entry on oeis.org

0, 1, 1, 3, 1, 6, 1, 7, 4, 8, 1, 16, 1, 10, 9, 15, 1, 21, 1, 22, 11, 14, 1, 24, 6, 16, 13, 28, 1, 42, 1, 31, 15, 20, 13, 25, 1, 22, 17, 30, 1, 54, 1, 40, 33, 26, 1, 64, 8, 43, 21, 46, 1, 48, 17, 64, 23, 32, 1, 46, 1, 34, 41, 63, 19, 78, 1, 58, 27, 74, 1, 93, 1

Offset: 1

Amiram Eldar, Aug 11 2023

nonn

First differs from A294888 at n = 48.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001065, A118372, A181487, A294888.

seq[nmax_] := Module[{s = {1}, a = {0}, sum}, Do[sum = Total[Select[Divisors[n], MemberQ[s, #] &]]; If[sum <= n, AppendTo[s, n]]; AppendTo[a, sum], {n, 2, nmax}]; a]; seq[100]

lista(nmax) = {my(c = 0, s); print1(0, ", "); for(n=2, nmax, s = sumdiv(n, d, !bittest(c, d)*d) - n; if(s > n, c+=1<M. F. Hasler at A181487

a(n) <= A001065(n).
a(n) <= n if and only if n is in the set S.
a(n) = n if and only if n is S-perfect (A118372).
a(n) > n if and only if n is S-abundant (A181487).

A364859 Lesser of a pair of S-amicable numbers k < m such that s(k) = m and s(m) = k, where s(k) = A364858(k).

Original entry on oeis.org

186, 1184, 2030, 6232, 10744, 66928, 522405, 643336, 5459176, 7677248, 13223490, 16137628, 25596544, 26090325, 28118032, 31772524, 34364912, 40504324, 133178325

Offset: 1

Amiram Eldar, Aug 11 2023

S-amicable numbers are analogous to amicable numbers (A002025/A002046) as S-perfect numbers (A118372) are analogous to perfect numbers (A000396).

The greater counterparts are in A364860.

			186 is a term since A364858(186) = 198 > 186, and A364858(198) = 186.

Cf. A000396, A002025, A002046, A118372, A364858, A364860.

seq[nmax_] := Module[{s = {1}, sum, sum2, am = {}, ak}, Do[sum = Total[Select[Divisors[n], MemberQ[s, #] &]]; If[sum <= n, AppendTo[s, n]; If[sum < n, sum2 = Total[Select[Most[Divisors[sum]], MemberQ[s, #] &]]; If[sum2 == n, AppendTo[am, sum]]]], {n, 2, nmax}]; am]; seq[10^4]

lista(nmax) = {my(c = 0, s, s2); for(n=2, nmax, s = sumdiv(n, d, !bittest(c, d)*d) - n; if(s > n, c+=1<M. F. Hasler at A181487

a(n) = A364858(A364860(n)).

A364860 Greater of a pair of S-amicable numbers k < m such that s(k) = m and s(m) = k, where s(k) = A364858(k).

Original entry on oeis.org

198, 1210, 2220, 6368, 10856, 66992, 525915, 652664, 5495264, 7684672, 13727466, 16150628, 25640096, 26138475, 28128368, 33642028, 34380688, 40803868, 133471275

Offset: 1

Amiram Eldar, Aug 11 2023

S-amicable numbers are analogous to amicable numbers (A002025/A002046) as S-perfect numbers (A118372) are analogous to perfect numbers (A000396).

The terms are ordered according to their lesser counterparts (A364859).

			198 is a term since A364858(198) = 186 < 198, and A364858(186) = 198.

Cf. A000396, A002025, A002046, A118372, A364858, A364859.

seq[nmax_] := Module[{s = {1}, sum, sum2, am = {}, ak}, Do[sum = Total[Select[Divisors[n], MemberQ[s, #] &]]; If[sum <= n, AppendTo[s, n]; If[sum < n, sum2 = Total[Select[Most[Divisors[sum]], MemberQ[s, #] &]]; If[sum2 == n, AppendTo[am, n]]]], {n, 2, nmax}]; am]; seq[10^4]

lista(nmax) = {my(c = 0, s, s2); for(n=2, nmax, s = sumdiv(n, d, !bittest(c, d)*d) - n; if(s > n, c+=1<M. F. Hasler at A181487

a(n) = A364858(A364859(n)).

cp's OEIS Frontend

A364861 Numbers k such that k and k+1 are both S-abundant numbers (A181487).

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A364862 S-weird numbers: S-abundant numbers (A181487) k such that no subset of the aliquot divisors of k that are in the set S sums to k, where S is the set defined in A118372.

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A118372 S-perfect numbers.

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A364858 a(n) = Sum_{d|n, d < n, d in S} d, where S is the set defined in A118372.

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A364859 Lesser of a pair of S-amicable numbers k < m such that s(k) = m and s(m) = k, where s(k) = A364858(k).

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A364860 Greater of a pair of S-amicable numbers k < m such that s(k) = m and s(m) = k, where s(k) = A364858(k).

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