A063990 -id:A063990 - cp's OEIS frontend

A127661 Lengths of the infinitary aliquot sequences.

2, 3, 3, 3, 3, 1, 3, 4, 3, 5, 3, 5, 3, 6, 4, 3, 3, 6, 3, 6, 4, 7, 3, 8, 3, 4, 4, 6, 3, 6, 3, 4, 5, 7, 4, 7, 3, 8, 4, 8, 3, 5, 3, 4, 5, 5, 3, 7, 3, 7, 5, 7, 3, 4, 4, 6, 4, 5, 3, 1, 3, 8, 4, 5, 4, 3, 3, 8, 5, 10, 3, 3, 3, 9, 4, 9, 4, 2, 3, 8, 3, 5, 3, 10, 4, 6, 6, 8, 3, 1, 5, 7, 5, 8, 4, 9, 3, 8, 5, 7

Offset: 1

Ant King, Jan 26 2007

nonn

An infinitary aliquot sequence is defined by the map x->A049417(x)-x. The map usually terminates with a zero, but may enter cycles (if n in A127662 for example).
The length of an infinitary aliquot sequence is defined to be the length of its transient part + the length of its terminal cycle.
The value of a(840) starting the infinitary aliquot sequence 840 -> 2040 -> 4440 -> 9240 -> 25320,... is >1500. - R. J. Mathar, Oct 05 2017

			a(4)=3 because the infinitary aliquot sequence generated by 4 is 4 -> 1 -> 0 and it has length 3.
a(6) = 1 because 6 -> 6 -> 6 ->... enters a cycle after 1 term.
a(8) = 4 because 8 -> 7 -> 1 -> 0 terminates after 4 terms.
a(30) = 6 because 30 ->42 -> 54 -> 66 -> 78 -> 90 -> 90 -> 90 -> ...enters a cycle after 6 terms.
a(126)=2 because 126 -> 114 -> 126 enters a cycle after 2 terms.

R. J. Mathar, Table of n, a(n) for n = 1..839
Graeme L. Cohen, On an integer's infinitary divisors, Math. Comp., 54 (1990), 395-411.
Hans Havermann, Graphs of infinitary aliquot sequences for 840, 1152, 2442, 2658, 2982, 5766, 6216, 6870, 7560, 8670, 9030, 9570 (click to see full plots)
D. Moews, A database of aliquot cycles (2015)
J. O. M. Pedersen, Tables of Aliquot Cycles [Broken link]
J. O. M. Pedersen, Tables of Aliquot Cycles [Via Internet Archive Wayback-Machine]
J. O. M. Pedersen, Tables of Aliquot Cycles [Cached copy, pdf file only]

Cf. A126168, A127662 - A127667, A293355, A098007.

# Uses code snippets of A049417
A127661 := proc(n)
    local trac,x;
    x := n ;
    trac := [x] ;
    while true do
        x := A049417(x)-trac[-1] ;
        if x = 0 then
            return 1+nops(trac) ;
        elif x in trac then
            return nops(trac) ;
        end if;
        trac := [op(trac),x] ;
    end do:
end proc:
seq(A127661(n),n=1..100) ; # R. J. Mathar, Oct 05 2017

ExponentList[n_Integer,factors_List]:={#,IntegerExponent[n,# ]}&/@factors;InfinitaryDivisors[1]:={1}; InfinitaryDivisors[n_Integer?Positive]:=Module[ { factors=First/@FactorInteger[n], d=Divisors[n] }, d[[Flatten[Position[ Transpose[ Thread[Function[{f,g}, BitOr[f,g]==g][ #,Last[ # ]]]&/@ Transpose[Last/@ExponentList[ #,factors]&/@d]],?(And@@#&),{1}]] ]] ]; properinfinitarydivisorsum[k]:=Plus@@InfinitaryDivisors[k]-k;g[n_] := If[n > 0,properinfinitarydivisorsum[n], 0];iTrajectory[n_] := Most[NestWhileList[g, n, UnsameQ, All]];Length[iTrajectory[ # ]] &/@ Range[100]
(* Second program: *)
A049417[n_] := If[n == 1, 1, Sort@ Flatten@ Outer[Times, Sequence @@ (FactorInteger[n] /. {p_, m_Integer} :> p^Select[Range[0, m], BitOr[m, #] == m &])]] // Total;
A127661[n_] := Module[{trac, x}, x = n; trac = {x}; While[True, x = A049417[x] - trac[[-1]]; If[x == 0, Return[1 + Length[trac]], If[MemberQ[trac, x], Return[Length[trac]]]]; trac = Append[trac, x]]];
Table[A127661[n], {n, 1, 100}] (* Jean-François Alcover, Aug 28 2023, after R. J. Mathar *)

A038148 Number of 3-infinitary divisors of n: if n = Product p(i)^r(i) and d = Product p(i)^s(i), each s(i) has a digit a <= b in its ternary expansion everywhere that the corresponding r(i) has a digit b, then d is a 3-infinitary-divisor of n.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 2, 3, 4, 2, 6, 2, 4, 4, 4, 2, 6, 2, 6, 4, 4, 2, 4, 3, 4, 2, 6, 2, 8, 2, 6, 4, 4, 4, 9, 2, 4, 4, 4, 2, 8, 2, 6, 6, 4, 2, 8, 3, 6, 4, 6, 2, 4, 4, 4, 4, 4, 2, 12, 2, 4, 6, 3, 4, 8, 2, 6, 4, 8, 2, 6, 2, 4, 6, 6, 4, 8, 2, 8, 4, 4, 2, 12, 4, 4, 4, 4, 2, 12, 4, 6, 4, 4, 4, 12, 2, 6, 6, 9, 2, 8, 2, 4, 8

Offset: 1

Yasutoshi Kohmoto

Multiplicative: If e = sum d_k 3^k, then a(p^e) = prod (d_k+1). - Christian G. Bower, May 19 2005

			2^3*3 is a 3-infinitary-divisor of 2^5*3^2 because 2^3*3 = 2^10*3^1 and 2^5*3^2 = 2^12*3^2 in ternary expanded power. All corresponding digits satisfy the condition. 1 <= 1, 0 <= 2, 1 <= 2.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Frédéric Chyzak, Ivan Gutman, and Peter Paule, Predicting the number of hexagonal systems with 24 and 25 hexagons, Communications in Mathematical and Computer Chemistry (1999) No. 40, 139-151. See p. 141.
J. O. M. Pedersen, Tables of Aliquot Cycles [Broken link]
J. O. M. Pedersen, Tables of Aliquot Cycles [Via Internet Archive Wayback-Machine]
J. O. M. Pedersen, Tables of Aliquot Cycles [Cached copy, pdf file only]
Index entries for sequences computed from exponents in factorization of n

Cf. A006047, A037445, A038182, A074848.

A038148 := proc(n) if n= 1 then 1; else ifa := ifactors(n)[2] ;
a := 1; for f in ifa do e := convert(op(2,f),base,3) ; a := a*mul(d+1,d=e) ; end do: end if; end proc:
seq(A038148(n),n=1..50) ; # R. J. Mathar, Feb 08 2011

a[1] = 1; a[n_] := (k = 1; Do[k = k * Times @@ (IntegerDigits[f, 3] + 1), {f, FactorInteger[n][[All, 2]]}]; k); Table[a[n], {n, 1, 102}](* Jean-François Alcover, Feb 03 2012, after R. J. Mathar *)

A006047(n) = { my(m=1, d); while(n, d = (n%3); m *= (1+d); n = (n-d)/3); m; };
A038148(n) = factorback(apply(e -> A006047(e), factorint(n)[, 2])); \\ (After A037445) - Antti Karttunen, May 28 2017

(define (A038148 n) (if (= 1 n) n (* (A006047 (A067029 n)) (A038148 (A028234 n))))) ;; Antti Karttunen, May 28 2017

a(1) = 1; for n > 1, a(n) = A006047(A067029(n)) * a(A028234(n)). [After Christian G. Bower's 2005 comment.] - Antti Karttunen, May 28 2017

More terms from Naohiro Nomoto, Jun 21 2001

Data section further extended to 105 terms by Antti Karttunen, May 28 2017

A121507 Conjectured list of numbers whose aliquot sequence eventually reaches a cycle of length two or more.

Original entry on oeis.org

220, 284, 562, 1064, 1184, 1188, 1210, 1308, 1336, 1380, 1420, 1490, 1604, 1690, 1692, 1772, 1816, 1898, 2008, 2122, 2152, 2172, 2362, 2542, 2620, 2630, 2652, 2676, 2678, 2856, 2924, 2930, 2950, 2974, 3124, 3162, 3202, 3278, 3286, 3332, 3350, 3360

Offset: 1

Joshua Zucker, Aug 04 2006

nonn

For some numbers the outcome of the aliquot sequence is unknown. Currently, 276 is the least such.

Cf. A008892, A063769, A063990, A072890, A072891, A098007, A121508.

Edited by Don Reble, Aug 15 2006

A262625 Odd amicable numbers.

Original entry on oeis.org

12285, 14595, 67095, 69615, 71145, 87633, 100485, 122265, 124155, 139815, 522405, 525915, 802725, 863835, 947835, 1125765, 1175265, 1280565, 1340235, 1358595, 1438983, 1486845, 1798875, 1870245, 4482765, 5120595, 5357625, 5684679, 5730615, 6088905, 6377175, 6680025, 8619765, 9071685, 9206925, 9491625, 9498555, 9627915

Offset: 1

Omar E. Pol, Oct 02 2015

nonn

Odd numbers that are also amicable numbers.

Intersection of A005408 and A063990.

Antti Karttunen, Table of n, a(n) for n = 1..30391 (odd numbers found from 77977-term b-file prepared for A063990 by _T. D. Noe_)

Cf. A005408, A063990, A262624.

A007992 Augmented amicable pairs (smaller member of each pair).

Original entry on oeis.org

6160, 12220, 23500, 68908, 249424, 425500, 434784, 649990, 660825, 1017856, 1077336, 1238380, 1252216, 1568260, 1754536, 2166136, 2362360, 2482536, 2537220, 2876445, 3957525, 4177524, 4287825, 5224660, 5559510, 5641552

Offset: 1

N. J. A. Sloane

Let f(n) = 1 + sum of aliquot divisors of n; these are pairs (n,m) with f(n)=m, f(m)=n.
m cannot equal n. - Harvey P. Dale, May 18 2012
The term "augmented amicable numbers" was coined by Beck and Wajar (1977), who found the first 11 pairs. They also found the next 25 pairs (1993). - Amiram Eldar, Mar 09 2024

Amiram Eldar, Table of n, a(n) for n = 1..1159
Walter E. Beck and Rudolph M. Wajar, More reduced amicable pairs, Fibonacci Quarterly, Vol. 15, No. 4 (1977), pp. 331-332.
Walter E. Beck and Rudolph M. Wajar, Reduced and Augmented Amicable Pairs to 10^8, Fibonacci Quarterly, Vol. 31, No. 4 (1993), pp. 295-298.
David Moews, Augmented amicable pairs.
J. O. M. Pedersen, Tables of Aliquot Cycles.
J. O. M. Pedersen, Tables of Aliquot Cycles. [Cached copy, pdf file only]
Paul Pollack, Quasi-Amicable Numbers are Rare, J. Int. Seq. 14 (2011), Article 11.5.2.
Eric Weisstein's World of Mathematics, Augmented Amicable Pair.

Cf. A015630.

aapQ[n_]:=Module[{c=DivisorSigma[1,n]+1-n},c!=n&&DivisorSigma[ 1,c]+1-c == n]; Transpose[Union[Sort[{#,DivisorSigma[1,#]+1-#}]&/@Select[Range[ 6000000],aapQ]]] [[1]] (* Harvey P. Dale, May 18 2012 *)

A015630 Augmented amicable pairs (larger member of each pair).

Original entry on oeis.org

11697, 16005, 28917, 76245, 339825, 570405, 871585, 697851, 678376, 1340865, 2067625, 1823925, 1483785, 1899261, 2479065, 2580105, 4895241, 4740505, 5736445, 3171556, 4791916, 6516237, 4416976, 7524525, 9868075, 7589745

Offset: 1

N. J. A. Sloane

Let f(n) = 1 + sum of aliquot divisors of n; these are pairs (n,m) with f(n)=m, f(m)=n.

The terms of the sequence are sorted in the order of the smaller (omitted) member of each pair. [Harvey P. Dale, Feb 29 2012]

Amiram Eldar, Table of n, a(n) for n = 1..1159
D. Moews, Augmented amicable pairs
J. O. M. Pedersen, Tables of Aliquot Cycles [Broken link]
J. O. M. Pedersen, Tables of Aliquot Cycles [Via Internet Archive Wayback-Machine]
J. O. M. Pedersen, Tables of Aliquot Cycles [Cached copy, pdf file only]
P. Pollack, Quasi-Amicable Numbers are Rare, J. Int. Seq. 14 (2011) # 11.5.2

Cf. A007992.

aap[n_]:=Module[{p=Total[Most[Divisors[n]]]+1},If[p!=n&&n==Total[Most[ Divisors[p]]]+1,{p,n},0]]; Transpose[Union[Sort/@DeleteCases[aap/@ Range[10000000],0]]][[2]] (* Harvey P. Dale, Feb 29 2012 *)

A049418 3-i-sigma(n): sum of 3-infinitary divisors of n: if n=Product p(i)^r(i) and d=Product p(i)^s(i), each s(i) has a digit a<=b in its ternary expansion everywhere that the corresponding r(i) has a digit b, then d is a 3-i-divisor of n.

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 9, 13, 18, 12, 28, 14, 24, 24, 27, 18, 39, 20, 42, 32, 36, 24, 36, 31, 42, 28, 56, 30, 72, 32, 63, 48, 54, 48, 91, 38, 60, 56, 54, 42, 96, 44, 84, 78, 72, 48, 108, 57, 93, 72, 98, 54, 84, 72, 72, 80, 90, 60, 168, 62, 96, 104, 73, 84, 144, 68, 126, 96

Offset: 1

Yasutoshi Kohmoto

			Let n = 28 = 2^2*7. Then a(n) = (2^2 + 2 + 1)*(7 + 1) = 56. - _Vladimir Shevelev_, May 07 2013

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
J. O. M. Pedersen, Tables of Aliquot Cycles, backup on web.archive.org of no more exiting page, as of May 2014
J. O. M. Pedersen, Tables of Aliquot Cycles [Cached copy, pdf file only]

Cf. A049417 (2-infinitary), A074847 (4-infinitary), A097863 (5-infinitary).

Cf. A000244, A030341, A027748, A124010.

following Bower and Harris:
a049418 1 = 1
a049418 n = product $ zipWith f (a027748_row n) (a124010_row n) where
   f p e = product $ zipWith div
           (map (subtract 1 . (p ^)) $
                zipWith (*) a000244_list $ map (+ 1) $ a030341_row e)
           (map (subtract 1 . (p ^)) a000244_list)
-- Reinhard Zumkeller, Sep 18 2015

A049418 := proc(n) option remember; local ifa,a,p,e,d,k ; ifa := ifactors(n)[2] ; a := 1 ; if nops(ifa) = 1 then p := op(1,op(1,ifa)) ; e := op(2,op(1,ifa)) ; d := convert(e,base,3) ; for k from 0 to nops(d)-1 do a := a*(p^((1+op(k+1,d))*3^k)-1)/(p^(3^k)-1) ; end do: else for d in ifa do a := a*procname( op(1,d)^op(2,d)) ; end do: return a; end if; end proc:
seq(A049418(n),n=1..40) ; # R. J. Mathar, Oct 06 2010

A049418[n_] := Module[{ifa = FactorInteger[n], a = 1, p, e, d, k}, If[ Length[ifa] == 1, p = ifa[[1, 1]]; e = ifa[[1, 2]]; d = Reverse[ IntegerDigits[e, 3] ]; For[k = 1, k <= Length[d], k++, a = a*(p^((1 + d[[k]])*3^(k - 1)) - 1)/(p^(3^(k - 1)) - 1)], Do[ a = a*A049418[ d[[1]]^d[[2]] ], {d, ifa}]]; Return[a] ]; A049418[1] = 1; Table[ A049418[n] , {n, 1, 69}] (* Jean-François Alcover, Jan 03 2012, after R. J. Mathar *)

apply( {A049418(n)=vecprod([prod(k=1,#n=digits(f[2],3),(f[1]^(3^(#n-k)*(n[k]+1))-1)\(f[1]^3^(#n-k)-1))|f<-factor(n)~])}, [1..99]) \\ M. F. Hasler, Sep 21 2022

Multiplicative with a(p^e) = prod_{k >= 0} (p^(3^k*{d_k+1}) - 1)/(p^(3^k) - 1), where e = sum_{k >= 0} d_k 3^k (base 3 representation). - Christian G. Bower and Mitch Harris, May 20 2005. [Edited by M. F. Hasler, Sep 21 2022]

Denote P_3 = {p^3^k}, k = 0, 1, ..., p runs primes. Then every n has a unique representation of the form n = prod q_i prod (r_j)^2, where q_i, r_j are distinct elements of P_3. Using this representation, we have a(n) = prod (q_i+1)*prod ((r_j)^2+r_j+1). - Vladimir Shevelev, May 07 2013

More terms from Naohiro Nomoto, Sep 10 2001

A122726 Conjectured list of sociable numbers.

Original entry on oeis.org

12496, 14264, 14288, 14316, 14536, 15472, 17716, 19116, 19916, 22744, 22976, 31704, 45946, 47616, 48976, 83328, 97946, 122410, 152990, 177792, 243760, 274924, 275444, 285778, 294896, 295488, 358336, 366556, 376736, 381028, 418904, 589786

Offset: 1

Tanya Khovanova, Sep 23 2006

nonn

Comments from David Moews, Sep 17 2021: (Start)
It is possible that there are quite small numbers missing from this sequence. There is no proof that 564 (for example) is missing.
Let s(n) = sigma(n)-n denote the sum of the divisors of n, excluding n itself. The aliquot sequence starting at n is the sequence n, s(n), s(s(n)), s(s(s(n))), ...
Starting at 564, the aliquot sequence continues for at least 3486 steps, reaching a 198-digit number after 3486 iterations of s.  In the reverse direction, 563 = 564 - 1 is prime so s(563^2) = 564 and also s(7*316961) = 563^2, s(17*2218709) = 7*316961, etc.; given a strengthened form of the Goldbach conjecture (see Booker, 2018), one can continue iterating s^(-1) indefinitely.
Although it seems unlikely, I don't see any way to be completely certain that the forward aliquot sequence doesn't meet the backwards tree; if it did, 564 would be part of a (very long) aliquot cycle.
Many other numbers below 79750 are in a similar situation (although not 276, because it is not in the image of s).
(Added Sep 18 2021) The smallest uncertain number is 564.  All smaller numbers either have known aliquot sequences (all except 276, 306, 396, and 552), are not in the image of s (276, 306, and 552), or are in the image of s but not the image of s^2 (396).
(End)

			The smallest sociable number cycle is {12496, 14288, 15472, 14536, 14264, 12496}.

Andrew R. Booker, Finite connected components of the aliquot graph, Math. Comp. 87 (2018), 2891-2902.
Steven Finch, Amicable Pairs and Aliquot Sequences, 2013. [Cached copy, with permission of the author]
David Moews, Perfect, amicable and sociable numbers
D. Moews, A list of currently known aliquot cycles of length greater than 2 [This list is not known to be complete.]
Eric Weisstein's World of Mathematics, Sociable Numbers.
Wikipedia, Sociable number

Cf. A003416 (smallest member of each cycle), A063990 (amicable numbers), A052470.

Edited (including adding comments from David Moews that this is only conjectural) by N. J. A. Sloane, Sep 17 2021

A275701 Numbers n whose abundance is 26: sigma(n) - 2n = 26.

Original entry on oeis.org

80, 1184, 6464, 29312, 78975, 510464, 557192, 137431875584, 549741658112, 8796036399104, 35184258842624, 2251798907715584

Offset: 1

Timothy L. Tiffin, Aug 05 2016

Any term x = a(m) can be combined with any term y = A275702(n) to satisfy the property (sigma(x)+sigma(y))/(x+y) = 2. Although this property is a necessary condition for two numbers to be amicable, it is not a sufficient one. So far, these two sequences have produced only one amicable pair: (x,y) = (1184,1210) = (a(2),A275702(5)) = (A063990(3),A063990(4)). If more are ever found, then they will also exhibit y-x = 26.
Notice that:
a(1) =     80 =   5* 16 = (2*4^2-27)*(4^2)
a(2) =   1184 =  37* 32 =   (4^3-27)*(4^3)/2
a(3) =   6464 = 101* 64 = (2*4^3-27)*(4^3)
a(4) =  29312 = 229*128 =   (4^4-27)*(4^4)/2
a(6) = 510464 = 997*512 =   (4^5-27)*(4^5)/2.
If p = 2*4^k-27 is prime and n = p*(p+27)/2, then it is not hard to show that sigma(n) - 2*n = 26. The values of k in A275767 will guarantee that p is prime (A275749). Similarly, if q = 4^k-27 is prime and n = q*(q+27)/2, then sigma(n) - 2*n = 26. The values of k in A274519 will guarantee that q is prime (A275750). So, the following values will be in this sequence and provide upper bounds for the next eight terms:
(2*4^9-27)*(4^9)     = 137431875584 >= a(8)
  (4^10-27)*(4^10)/2 = 549741658112 >= a(9)
  (4^11-27)*(4^11)/2 = 8796036399104 >= a(10)
(2*4^11-27)*(4^11)   = 35184258842624 >= a(11)
  (4^13-27)*(4^13)/2 = 2251798907715584 >= a(12)
  (4^25-27)*(4^25)/2 = 633825300114099501099609227264 >= a(13)
  (4^28-27)*(4^28)/2 = 2596148429267412841487728652582912 >= a(14)
  (4^29-27)*(4^29)/2 = 41538374868278617137133892585652224 >= a(15).
a(8) > 10^9. - Michel Marcus, Sep 15 2016
a(8) > 2*10^9. - Michel Marcus, Dec 31 2016
a(13) > 10^18. - Hiroaki Yamanouchi, Aug 23 2018

			a(1) = 80, since sigma(80)-2*80 = 186-160 = 26.
a(2) = 1184, since sigma(1184)-2*1184 = 2394-2368 = 26.
a(3) = 6464, since sigma(6464)-2*6464 = 12954-12928 = 26.

D. Alpern, Factorization using the Elliptic Curve Method.

Cf. A033880, A063990, A274519, A275702 (deficiency 26), A275749, A275750, A275767.

Cf. A223609 (abundance 10), ..., A223613 (abundance 24).

[n: n in [1..9*10^6] | (SumOfDivisors(n)-2*n) eq 26]; // Vincenzo Librandi, Sep 16 2016

Select[Range[10^7], DivisorSigma[1, #] - 2 # == 26 &] (* Vincenzo Librandi, Sep 16 2016 *)

isok(n) = sigma(n) - 2*n == 26; \\ Michel Marcus, Sep 15 2016

a(8)-a(12) from Hiroaki Yamanouchi, Aug 23 2018

A260086 Smaller of amicable pair (x, y) as they are listed in A259933.

Original entry on oeis.org

220, 1184, 2620, 5020, 6232, 10744, 12285, 17296, 66928, 67095, 63020, 69615, 79750, 100485, 122368, 122265, 141664, 142310, 171856, 176272, 185368, 196724, 280540, 308620, 319950, 356408, 437456, 469028, 503056, 522405, 600392, 609928, 643336, 624184, 635624, 667964, 726104, 802725, 879712, 898216, 998104, 947835

Offset: 1

Omar E. Pol, Jul 15 2015

nonn

Another version of A002025.

First differs from A002025 at a(9).

Laszlo Hars, Performance compared to mathematica Julia-users (2014)
Khelleos, Amicable numbers, CyberForum.ru (2011)
OEIS Wiki, Amicable numbers (This page needs work)
Wikipédia, Barátságos számok (contains a mistake: A063990 should be replaced with A259933)

Cf. A002025, A063990, A259180, A259933, A259953, A260087.

a(n) = A259933(2n-1) = A259953(n) - A259933(2n) = A259953(n) - A260087(n).

cp's OEIS Frontend

A127661 Lengths of the infinitary aliquot sequences.

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A038148 Number of 3-infinitary divisors of n: if n = Product p(i)^r(i) and d = Product p(i)^s(i), each s(i) has a digit a <= b in its ternary expansion everywhere that the corresponding r(i) has a digit b, then d is a 3-infinitary-divisor of n.

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A121507 Conjectured list of numbers whose aliquot sequence eventually reaches a cycle of length two or more.

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A262625 Odd amicable numbers.

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A007992 Augmented amicable pairs (smaller member of each pair).

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A015630 Augmented amicable pairs (larger member of each pair).

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A049418 3-i-sigma(n): sum of 3-infinitary divisors of n: if n=Product p(i)^r(i) and d=Product p(i)^s(i), each s(i) has a digit a<=b in its ternary expansion everywhere that the corresponding r(i) has a digit b, then d is a 3-i-divisor of n.

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A122726 Conjectured list of sociable numbers.

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