A127661 -id:A127661 - cp's OEIS frontend

A049417 a(n) = isigma(n): sum of infinitary divisors of n.

1, 3, 4, 5, 6, 12, 8, 15, 10, 18, 12, 20, 14, 24, 24, 17, 18, 30, 20, 30, 32, 36, 24, 60, 26, 42, 40, 40, 30, 72, 32, 51, 48, 54, 48, 50, 38, 60, 56, 90, 42, 96, 44, 60, 60, 72, 48, 68, 50, 78, 72, 70, 54, 120, 72, 120, 80, 90, 60, 120, 62, 96, 80, 85, 84, 144, 68, 90

Offset: 1

Yasutoshi Kohmoto, Dec 11 1999

A divisor of n is called infinitary if it is a product of divisors of the form p^{y_a 2^a}, where p^y is a prime power dividing n and sum_a y_a 2^a is the binary representation of y.
This sequence is an infinitary analog of the Dedekind psi function A001615. Indeed, a(n) = Product_{q in Q_n}(q+1) =  n*Product_{q in Q_n} (1+1/q), where {q} are terms of A050376 and Q_n is the set of distinct q's whose product is n. - Vladimir Shevelev, Apr 01 2014
1/a(n) is the asymptotic density of numbers that are infinitarily divided by n (i.e., numbers whose set of infinitary divisors includes n). - Amiram Eldar, Jul 23 2025

			If n = 8: 8 = 2^3 = 2^"11" (writing 3 in binary) so the infinitary divisors are 2^"00" = 1, 2^"01" = 2, 2^"10" = 4 and 2^"11" = 8; so a(8) = 1+2+4+8 = 15.
n = 90 = 2*5*9, where 2, 5, 9 are in A050376; so a(n) = 3*6*10 = 180. - _Vladimir Shevelev_, Feb 19 2011

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..7417 from R. J. Mathar)
Graeme L. Cohen, On an integer's infinitary divisors, Math. Comp. 54 (189) (1990) 395-411.
Steven R. Finch, Unitarism and Infinitarism, February 25, 2004. [Cached copy, with permission of the author]
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, pp. 49-56.
J. O. M. Pedersen, Tables of Aliquot Cycles.
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]
Tomohiro Yamada, Infinitary superperfect numbers, Annales Mathematicae et Informaticae, Vol. 47 (2017), pp. 211-218; arXiv preprint, arXiv:1705.10933 [math.NT], 2017.

Cf. A004607, A037445, A050376, A077609, A091732, A127661, A293355.
Cf. A049418 (3-infinitary), A074847 (4-infinitary), A097863 (5-infinitary).
Cf. A000079, A001615, A027748, A030308, A124010, A126168.

a049417 1 = 1
a049417 n = product $ zipWith f (a027748_row n) (a124010_row n) where
   f p e = product $ zipWith div
           (map (subtract 1 . (p ^)) $
                zipWith (*) a000079_list $ map (+ 1) $ a030308_row e)
           (map (subtract 1 . (p ^)) a000079_list)
-- Reinhard Zumkeller, Sep 18 2015

isidiv := proc(d, n)
    local n2, d2, p, j;
    if n mod d <> 0 then
        return false;
    end if;
    for p in numtheory[factorset](n) do
        padic[ordp](n,p) ;
        n2 := convert(%, base, 2) ;
        padic[ordp](d,p) ;
        d2 := convert(%, base, 2) ;
        for j from 1 to nops(d2) do
            if op(j, n2) = 0 and op(j, d2) <> 0 then
                return false;
            end if;
        end do:
    end do;
    return true;
end proc:
idivisors := proc(n)
    local a, d;
    a := {} ;
    for d in numtheory[divisors](n) do
        if isidiv(d, n) then
            a := a union {d} ;
        end if;
    end do:
    a ;
end proc:
A049417 := proc(n)
    local d;
    add(d, d=idivisors(n)) ;
end proc:
seq(A049417(n),n=1..100) ; # R. J. Mathar, Feb 19 2011

bitty[k_] := Union[Flatten[Outer[Plus, Sequence @@ ({0, #1} & ) /@ Union[2^Range[0, Floor[Log[2, k]]]*Reverse[IntegerDigits[k, 2]]]]]]; Table[Plus@@((Times @@ (First[it]^(#1 /. z -> List)) & ) /@ Flatten[Outer[z, Sequence @@ bitty /@ Last[it = Transpose[FactorInteger[k]]], 1]]), {k, 2, 120}]
(* Second program: *)
a[n_] := If[n == 1, 1, Sort @ Flatten @ Outer[ Times, Sequence @@ (FactorInteger[n] /. {p_, m_Integer} :> p^Select[Range[0, m], BitOr[m, #] == m &])]] // Total;
Array[a, 100] (* Jean-François Alcover, Mar 23 2020, after Paul Abbott in A077609 *)

A049417(n) = {my(b, f=factorint(n)); prod(k=1, #f[,2], b = binary(f[k,2]); prod(j=1, #b, if(b[j], 1+f[k,1]^(2^(#b-j)), 1)))} \\ Andrew Lelechenko, Apr 22 2014

isigma(n)=vecprod([vecprod([f[1]^2^k+1|k<-[0..exponent(f[2])], bittest(f[2],k)])|f<-factor(n)~]) \\ M. F. Hasler, Oct 20 2022

from math import prod
from sympy import factorint
def A049417(n): return prod(p**(1<Chai Wah Wu, Jul 11 2024

Multiplicative: If e = Sum_{k >= 0} d_k 2^k (binary representation of e), then a(p^e) = Product_{k >= 0} (p^(2^k*{d_k+1}) - 1)/(p^(2^k) - 1). - Christian G. Bower and Mitch Harris, May 20 2005 [This means there is a factor p^2^k + 1 if d_k = 1, otherwise the factor is 1. - M. F. Hasler, Oct 20 2022]
Let n = Product(q_i) where {q_i} is a set of distinct terms of A050376. Then a(n) = Product(q_i + 1). - Vladimir Shevelev, Feb 19 2011
If n is squarefree, then a(n) = A001615(n). - Vladimir Shevelev, Apr 01 2014
a(n) = Sum_{k>=1} A077609(n,k). - R. J. Mathar, Oct 04 2017
a(n) = A126168(n)+n. - R. J. Mathar, Oct 05 2017
Multiplicative with a(p^e) = Product{k >= 0, e_k = 1} p^2^k + 1, where e = Sum e_k 2^k, i.e., e_k is bit k of e. - M. F. Hasler, Oct 20 2022
a(n) = iphi(n^2)/iphi(n), where iphi(n) = A091732(n). - Amiram Eldar, Sep 21 2024

More terms from Wouter Meeussen, Sep 02 2001

A127666 Odd infinitary abundant numbers.

Original entry on oeis.org

945, 10395, 12285, 15015, 16065, 17955, 19305, 19635, 21735, 21945, 23205, 23625, 25245, 25935, 26565, 27405, 28215, 28875, 29295, 29835, 31395, 33345, 33495, 33915, 34125, 34155, 34965, 35805, 37125, 38745, 39585, 40635, 41055, 42315

Offset: 1

Ant King, Jan 26 2007

nonn

This is also the sequence of odd integers whose infinitary aliquot sequences initially increase. Based on empirical evidence (up to 10 million), this applies to only about 0.1% of odd integers.

The numbers of terms not exceeding 10^k, for k = 4, 5, ..., are 1, 77, 473, 5703, 53569, 561610, 5525461, 54979537, ... . Apparently, the asymptotic density of this sequence exists and equals 0.0005... . - Amiram Eldar, Sep 09 2022

			a(5)=16065 because 16065 is the fifth odd number that is exceeded by the sum of its proper infinitary divisors.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Graeme L. Cohen, On an integer's infinitary divisors, Math. Comp., Vol. 54, No. 189, (1990), 395-411.
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. A005231, A126168, A127661, A129656.

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}]] ]] ] Null;properinfinitarydivisorsum[k]:=Plus@@InfinitaryDivisors[k]-k;Select[Range[1,50000,2],properinfinitarydivisorsum[ # ]># &] (* end of program *)
fun[p_, e_] := Module[{b = IntegerDigits[e, 2]}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]]; isigma[1] = 1; isigma[n_] := Times @@ fun @@@ FactorInteger[n]; Select[Range[1, 50000, 2], isigma[#] > 2 # &] (* Amiram Eldar, Jun 09 2019 *)

A049417(n) = {my(b, f=factorint(n)); prod(k=1, #f[, 2], b = binary(f[k, 2]); prod(j=1, #b, if(b[j], 1+f[k, 1]^(2^(#b-j)), 1)))}
isok(k) = A049417(k)>2*k&&k%2==1; \\ Jinyuan Wang, Jun 09 2019

Odd values of n for which A126168(n)>n.

A127667 Odd integers that do not generate monotonically decreasing infinitary aliquot sequences.

Original entry on oeis.org

945, 1743, 2175, 2655, 2823, 2865, 3105, 3375, 3537, 3585, 3729, 4209, 4665, 5775, 6559, 6681, 6969, 7257, 7263, 7785, 8457, 8583, 9657, 10017, 10047, 10113, 10395, 10599, 10743, 12285, 13815, 14055, 14145, 15015, 15597, 16065, 17955, 18529, 18777, 19305, 19635

Offset: 1

Ant King, Jan 26 2007

nonn

Based on empirical evidence, approximately 98.9 % of the infinitary aliquot sequences generated by the odd integers are monotonically decreasing. This sequence represents the 1.1 % of odd integers that are the exceptions to this.

			a(5)=2823 because 2823 is the fifth odd integer whose infinitary aliquot sequence is not monotonically decreasing.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Graeme L. Cohen, On an integer's infinitary divisors, Math. Comp., 54 (1990), 395-411.
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, A127661, A127666.

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}]] ]] ] Null;properinfinitarydivisorsum[k]:=Plus@@InfinitaryDivisors[k]-k;g[n_] := If[n > 0,properinfinitarydivisorsum[n], 0];iTrajectory[n_] := Most[NestWhileList[g, n, UnsameQ, All]];u[n_]:=Table[n[[k+1]]

More terms from Amiram Eldar, Sep 16 2019

A127656 Lengths of the exponential aliquot sequences.

Original entry on oeis.org

2, 2, 2, 3, 2, 2, 2, 3, 3, 2, 2, 3, 2, 2, 2, 3, 2, 3, 2, 3, 2, 2, 2, 3, 3, 2, 3, 3, 2, 2, 2, 3, 2, 2, 2, 1, 2, 2, 2, 3, 2, 2, 2, 3, 3, 2, 2, 4, 3, 3, 2, 3, 2, 3, 2, 3, 2, 2, 2, 3, 2, 2, 3, 3, 2, 2, 2, 3, 2, 2, 2, 5, 2, 2, 3, 3, 2, 2, 2, 3, 4, 2, 2, 3, 2, 2, 2, 3, 2, 3, 2, 3, 2, 2, 2, 3, 2, 3, 3, 4

Offset: 1

Ant King, Jan 25 2007

nonn

The exponential aliquot sequence is defined by the map x -> A051377(x)-x starting at n.

The length of an exponential aliquot sequence is defined according to the length of its transient part + the length of its terminal cycle.

			a(4)=3 because the exponential aliquot sequence generated by 4 is <4,2,0> and it has length 3.
From _R. J. Mathar_, Oct 05 2017: (Start)
The aliquot sequnence may enter a cycle (see A054979)
36 -> 36 -> ..
180 -> 180 -> ..
252 -> 252 -> ..
396 -> 396 -> ..
468 -> 468 -> ..
612 -> 612 -> ..
684 -> 684 -> ..
828 -> 828 -> ..
900 -> 1260 -> 1260 -> ..
1044 -> 1044 -> ..
1116 -> 1116 -> ..
1260 -> 1260 -> ..
1332 -> 1332 -> ..
1352 -> 468 -> 468 -> ..
1476 -> 1476 -> ..
1548 -> 1548 -> ..
1692 -> 1692 -> ..
1728 -> 612 -> 612 -> ..
1800 -> 1800 -> ..
1908 -> 1908 -> ..
1980 -> 1980 -> ..
2124 -> 2124 -> ..
2196 -> 2196 -> ..
2340 -> 2340 -> ..
2412 -> 2412 -> ..
2556 -> 2556 -> ..
2628 -> 2628 -> ..
2700 -> 2700 -> ..
2772 -> 2772 -> ..
2844 -> 2844 -> ..
2880 -> 1800 -> 1800 -> ..
(End)

Hans Havermann, Table of n, a(n) for n = 1..10000
Hagis, Peter Jr., Some Results Concerning Exponential Divisors, Internat. J. Math. & Math. Sci., Vol. 11, No. 2, (1988), pp. 343-350.
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. A127657, A127658, A127659, A127660.

Cf. also A127661.

A127656 := proc(n)
    local trac,x;
    x := n ;
    trac := [x] ;
    while true do
        x := A051377(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: # R. J. Mathar, Oct 05 2017

ExponentialDivisors[1]={1};ExponentialDivisors[n_]:=Module[{}, {pr,pows}=Transpose@FactorInteger[n]; divpowers=Distribute[Divisors[pows],List];Sort[Times@@(pr^Transpose[divpowers])]];se[n_]:=Plus@@ExponentialDivisors[n]-n;g[n_] := If[n > 0, se[n], 0];eTrajectory[n_] := Most[NestWhileList[g, n, UnsameQ, All]];Length[eTrajectory[ # ]] &/@Range[100]
(* Second program: *)
f[n_] := Times @@ (Sum[First[#]^d, {d, Divisors[Last[#]]}]&) /@ FactorInteger[n];
a[n_] := Length[FixedPointList[f[#]-#&, n]]-1;
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jun 04 2023 *)

A127662 Integers whose infinitary aliquot sequences end in an infinitary perfect number (A007357).

Original entry on oeis.org

6, 30, 42, 54, 60, 66, 72, 78, 90, 100, 140, 148, 152, 192, 194, 196, 208, 220, 238, 244, 252, 268, 274, 292, 296, 298, 300, 336, 348, 350, 360, 364, 372, 374, 380, 382, 386, 400, 416, 420, 424, 476, 482, 492, 516, 520, 532, 540, 542, 544, 550, 572, 576, 578, 586

Offset: 1

Ant King, Jan 26 2007

			a(5) = 60 because the fifth number whose infinitary aliquot sequence ends in an infinitary perfect number is 60.
6 -> 6 ...
30 -> 42 -> 54 -> 66 -> 78 -> 90 -> 90 -> ..
42 -> 54 -> 66 -> 78 -> 90 -> 90 -> ..
54 -> 66 -> 78 -> 90 -> 90 -> ..
60 -> 60 -> ..
66 -> 78 -> 90 -> 90 -> ..
72 -> 78 -> 90 -> 90 -> ..
78 -> 90 -> 90 -> ..
90 -> 90 -> ..
100 -> 30 -> 42 -> 54 -> 66 -> 78 -> 90 -> 90 -> ..
102 -> 114 -> 126 -> 114 -> ..  cycle but not in the sequence
114 -> 126 -> 114 -> .. cycle but not in the sequence
126 -> 114 -> 126 -> ..
140 -> 100 -> 30 -> 42 -> 54 -> 66 -> 78 -> 90 -> 90 -> ..
148 -> 42 -> 54 -> 66 -> 78 -> 90 -> 90 -> ..
152 -> 148 -> 42 -> 54 -> 66 -> 78 -> 90 -> 90 -> ..
192 -> 148 -> 42 -> 54 -> 66 -> 78 -> 90 -> 90 -> ..
194 -> 100 -> 30 -> 42 -> 54 -> 66 -> 78 -> 90 -> 90 -> ..
196 -> 54 -> 66 -> 78 -> 90 -> 90 -> ..
208 -> 30 -> 42 -> 54 -> 66 -> 78 -> 90 -> 90 -> ..
210 -> 366 -> 378 -> 582 -> 594 -> 846 -> 594 -> ..
220 -> 140 -> 100 -> 30 -> 42 -> 54 -> 66 -> 78 -> 90 -> 90 -> ..
238 -> 194 -> 100 -> 30 -> 42 -> 54 -> 66 -> 78 -> 90 -> 90 -> ..
244 -> 66 -> 78 -> 90 -> 90 -> ..
246 -> 258 -> 270 -> 450 -> 330 -> 534 -> 546 -> 798 -> 1122 -> 1470 -> 2130 -> 3054 -> 3066 -> 4038 -> 4050 -> 2346 -> 2838 -> 3498 -> 4278 -> 4938 -> 4950 -> 4410 -> 4590 -> 8370 -> 14670 -> 14850 -> 22590 -> 22770 -> 29070 -> 35730 -> 35910 -> 79290 -> 79470 -> 79650 -> 107550 -> 79650 -> ..

Amiram Eldar, Table of n, a(n) for n = 1..75
Graeme L. Cohen, On an integer's infinitary divisors, Math. Comp., 54 (1990), 395-411.
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. A007357, A126168, A127661 - A127667.

isA007357 := proc(n)
    A049417(n) = 2*n ;
    simplify(%) ;
end proc:
isA127662 := proc(n)
    local trac,x;
    x := n ;
    trac := [x] ;
    while true do
        x := A049417(x)-trac[-1] ;
        if x = 0 then
            return false ;
        elif x in trac then
            return isA007357(x) ;
        end if;
        trac := [op(trac),x] ;
    end do:
end proc:
for n from 1 do
    if isA127662(n) then
        printf("%d,\n",n) ;
    end if;
end do: # 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}]] ]] ] Null;properinfinitarydivisorsum[k]:=Plus@@InfinitaryDivisors[k]-k;g[n_] := If[n > 0,properinfinitarydivisorsum[n], 0];iTrajectory[n_] := Most[NestWhileList[g, n, UnsameQ, All]];InfinitaryPerfectNumberQ[0]=False;InfinitaryPerfectNumberQ[k_Integer] :=If[properinfinitarydivisorsum[k]==k,True,False];Select[Range[500],InfinitaryPerfectNumberQ[Last[iTrajectory[ # ]]] &]
s[n_] := Times @@ (1 + Power @@@ FactorInteger[n]) - n; s[0] = s[1] = 0; q[n_] := Module[{v = NestWhileList[s, n, UnsameQ, All]}, v[[-1]] != n && v[[-2]] == v[[-1]] > 0]; Select[Range[3200], q] (* Amiram Eldar, Mar 11 2023 *)

More terms from Amiram Eldar, Mar 11 2023

A293355 Infinitary aliquot sequence starting at 6216.

Original entry on oeis.org

6216, 12024, 13176, 24024, 56616, 105624, 96096, 178080, 350688, 454704, 189528, 296472, 512808, 826392, 1535208, 2387352, 4124328, 6980472, 13561128, 29494872, 38242728, 42031872, 706200, 1315560, 2846040, 5936520, 12187320, 24375000, 57656040, 116396760, 233580840, 467162040

Offset: 0

Hans Havermann, Oct 07 2017

nonn

Is this sequence infinite?

The sequence reaches a local maximum at a(398), then descends to a local minimum at a(923) with a value of 20989279721046. Thereafter it rises again somewhat slowly and chaotically. - Hans Havermann, Jul 17 2021

Ray Chandler and Hans Havermann, Table of n, a(n) for n = 0..6042
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)
Hans Havermann, How the author first extended the b-file
Hans Havermann, Graph of a(n+1)/a(n)

Cf. A037445, A049417, A126168, A127661.

a(n+1) = A126168(a(n)). - R. J. Mathar, Oct 11 2017

A127663 Infinitary aspiring numbers.

Original entry on oeis.org

30, 42, 54, 66, 72, 78, 100, 140, 148, 152, 192, 194, 196, 208, 220, 238, 244, 252, 268, 274, 292, 296, 298, 300, 336, 348, 350, 360, 364, 372, 374, 380, 382, 386, 400, 416, 420, 424, 476, 482, 492, 516, 520, 532, 540, 542, 544, 550, 572, 576, 578, 586, 592

Offset: 1

Ant King, Jan 26 2007

Numbers whose infinitary aliquot sequences end in an infinitary perfect number, but are not infinitary perfect numbers themselves.

			a(5) = 72 because the fifth non-infinitary perfect number whose infinitary aliquot sequence ends in an infinitary perfect number is 72.

Amiram Eldar, Table of n, a(n) for n = 1..72
Graeme L. Cohen, On an integer's infinitary divisors, Math. Comp., 54 (1990), 395-411.
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. A007357, A126168, A127661, A127662, A127664, A127665, A127666, A127667.

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}]] ]] ] Null;properinfinitarydivisorsum[k]:=Plus@@InfinitaryDivisors[k]-k;g[n_] := If[n > 0,properinfinitarydivisorsum[n], 0];iTrajectory[n_] := Most[NestWhileList[g, n, UnsameQ, All]];InfinitaryPerfectNumberQ[0]=False;InfinitaryPerfectNumberQ[k_Integer] :=If[properinfinitarydivisorsum[k]==k,True,False];Select[Range[750],InfinitaryPerfectNumberQ[Last[iTrajectory[ # ]]] && !InfinitaryPerfectNumberQ[ # ]&]
f[p_, e_] := Module[{b = IntegerDigits[e, 2]}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]]; s[n_] := Times @@ f @@@ FactorInteger[n] - n; s[0] = s[1] = 0; q[n_] := Module[{v = NestWhileList[s, n, UnsameQ, All]}, n != v[[-2]] == v[[-1]] > 0]; Select[Range[839], q]  (* Amiram Eldar, Mar 11 2023 *)

A127665 Numbers whose infinitary aliquot sequences end in an infinitary amicable pair.

Original entry on oeis.org

102, 114, 126, 210, 246, 258, 270, 318, 330, 342, 354, 366, 378, 388, 390, 408, 426, 436, 438, 450, 474, 484, 486, 498, 510, 522, 534, 536, 546, 552, 570, 582, 594, 600, 606, 618, 630, 642, 648, 654, 666, 672, 702, 726, 738, 750, 760, 762, 774, 786, 798

Offset: 1

Ant King, Jan 26 2007

Sometimes called the infinitary 2-cycle attractor set.

			a(5)=246 because 246 is the fifth number whose infinitary aliquot sequence ends in an infinitary amicable pair.

Graeme L. Cohen, On an integer's infinitary divisors, Math. Comp., 54 (1990), 395-411.
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, A007357, A127661, A127662, A127663, A127664, A127666, A127667, A126169, A126170, A126171.

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}]] ]] ] Null;properinfinitarydivisorsum[k]:=Plus@@InfinitaryDivisors[k]-k;g[n_] := If[n > 0,properinfinitarydivisorsum[n], 0];iTrajectory[n_] := Most[NestWhileList[g, n, UnsameQ, All]];InfinitaryAmicableNumberQ[k_]:=If[Nest[properinfinitarydivisorsum,k,2]==k && !properinfinitarydivisorsum[k]==k,True,False];Select[Range[820],InfinitaryAmicableNumberQ[Last[iTrajectory[ # ]]] &]

A361421 Infinitary aliquot sequence starting at 840: a(1) = 840, a(n) = A126168(a(n-1)), for n >= 2.

Original entry on oeis.org

840, 2040, 4440, 9240, 25320, 51000, 117480, 271320, 765480, 1531320, 3721800, 5956440, 12295560, 25086840, 54141960, 108284280, 250301640, 502213560, 1007626440, 2017856760, 4039750920, 8079502200, 19596145800, 44369345400, 71495068200, 115576350360, 231152701080

Offset: 1

Amiram Eldar, Mar 11 2023

nonn

First differs from A045477 at n = 12.
840 is the least number whose infinitary aliquot sequence is not known to be finite or eventually periodic.
R. J. Mathar found that this sequence does not reach 0 or enter a cycle before the 1500th term (see A127661). This limit was extended to beyond the 3000th term (see the b-file).

			a(1) = 840 by definition.
a(2) = A126168(a(1)) = A126168(840) = 2040.
a(3) = A126168(a(2)) = A126168(2040) = 4440.

Amiram Eldar, Table of n, a(n) for n = 1..3132

Cf. A008892, A045477, A126168, A127661, A293355.

f[p_, e_] := Module[{b = IntegerDigits[e, 2]}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]];
infs[n_] := If[n==1, 1, Times @@ f @@@ FactorInteger[n]] - n; infs[0] = 0;
seq[len_, init_] := NestWhileList[infs, init, UnsameQ, All, len];
seq[27, 840]

s(n) = {my(f = factor(n), b); prod(i=1, #f~, b = binary(f[i, 2]); prod(k=1, #b, if(b[k], f[i, 1]^(2^(#b-k)) + 1, 1))) - n; }
lista(nmax) = {my(k = 840); for(n = 1, nmax, print1(k, ", "); if(k == 0, break); k = s(k)); }

A127664 Infinitary amicable numbers.

Original entry on oeis.org

114, 126, 594, 846, 1140, 1260, 4320, 5940, 7920, 8460, 8640, 10744, 10856, 11760, 12285, 13500, 14595, 17700, 25728, 35712, 43632, 44772, 45888, 49308, 60858, 62100, 62700, 67095, 67158, 71145, 73962, 74784, 79296, 79650, 79750, 83142, 83904, 86400, 88730

Offset: 1

Ant King, Jan 26 2007

nonn

			a(5)=1140 because 1140 is the fifth infinitary amicable number.

Graeme L. Cohen, On an integer's infinitary divisors, Math. Comp., 54 (1990), 395-411.
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. A007357, A126168, A127661, A127662, A127663, A127665, A127666, A127667, A126169, A126170, A126171.

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}]] ]] ] Null;properinfinitarydivisorsum[k]:=Plus@@InfinitaryDivisors[k]-k;g[n_] := If[n > 0,properinfinitarydivisorsum[n], 0];iTrajectory[n_] := Most[NestWhileList[g, n, UnsameQ, All]];InfinitaryAmicableNumberQ[k_]:=If[Nest[properinfinitarydivisorsum,k,2]==k && !properinfinitarydivisorsum[k]==k,True,False];Select[Range[50000],InfinitaryAmicableNumberQ[ # ] &]
fun[p_, e_] := Module[{b = IntegerDigits[e, 2]}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]]; infs[n_] := Times @@ (fun @@@ FactorInteger[n]) - n; s = {}; Do[k = infs[n]; If[k != n && infs[k] == n, AppendTo[s, n]], {n, 2, 10^5}]; s (* Amiram Eldar, Mar 16 2019 *)

Non-infinitary perfect numbers which satisfy A126168(A126168(n)) = n.

More terms from Amiram Eldar, Mar 16 2019

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