A127659 -id:A127659 - cp's OEIS frontend

A127660 Integers whose exponential aliquot sequences end in an exponential amicable pair.

90972, 100548, 454860, 502740, 937692, 968436, 1000692, 1106028, 1182636, 1307124, 1383732, 1536416, 1546524, 1709316, 2092356, 2312604, 2502528, 2638188, 2690100, 2820132, 2915892, 3116988, 3365964, 3720276, 3729852, 3907008, 3911796, 4122468, 4248552, 4275684

Offset: 1

Ant King, Jan 25 2007

nonn

Sometimes called the exponential 2-cycle attractor set. The first 10 terms of this sequence are the same as the first 10 terms of A127659.

			a(11) = 1383732 because the eleventh integer whose exponential aliquot sequence ends in an exponential amicable pair is 1383732.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Peter Hagis, 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. A127656, A127657, A127658.

Subsequences: A127659, A126165, A126166.

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]];ExponentialAmicableNumberQ[k_]:=If[Nest[se,k,2]==k && !se[k]==k,True,False];Select[Range[5 10^6],ExponentialAmicableNumberQ[Last[eTrajectory[ # ]]] &]
f[p_, e_] := DivisorSum[e, p^# &]; s[0] = s[1] = 0; s[n_] := Times @@ f @@@ FactorInteger[n] - n; q[n_] := Module[{v = NestWhileList[s, n, UnsameQ, All]}, v[[-2]] != v[[-1]] > 0 && v[[-3]] == v[[-1]]]; Select[Range[10^6], q] (* Amiram Eldar, Mar 11 2023 *)

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 *)

A127657 Integers whose exponential aliquot sequences end in an e-perfect number.

Original entry on oeis.org

36, 180, 252, 396, 468, 612, 684, 828, 900, 1044, 1116, 1260, 1332, 1352, 1476, 1548, 1692, 1728, 1800, 1908, 1980, 2124, 2196, 2340, 2412, 2556, 2628, 2700, 2772, 2844, 2880, 2916, 2988, 3000, 3060, 3204, 3276, 3420, 3492, 3636, 3708, 3750, 3852, 3924, 4068, 4140

Offset: 1

Ant King, Jan 25 2007

nonn

			a(5) = 468 because the fifth integer whose exponential aliquot sequences ends in an e-perfect number is 468.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Peter Hagis, 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. A127656, A127658, A127659, A127660, A054979.

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]];ExponentialPerfectNumberQ[0]=False;ExponentialPerfectNumberQ[k_Integer] :=If[se[k]==k,True,False];Select[Range[5000],ExponentialPerfectNumberQ[Last[eTrajectory[ # ]]] &]
f[p_, e_] := DivisorSum[e, p^# &]; s[0] = s[1] = 0; s[n_] := Times @@ f @@@ FactorInteger[n] - n; q[n_] := Module[{v = NestWhileList[s, n, UnsameQ, All]}, v[[-1]] == v[[-2]] > 0]; Select[Range[4000], q] (* Amiram Eldar, Mar 11 2023 *)

A127658 Exponential aspiring numbers.

Original entry on oeis.org

900, 1352, 1728, 2880, 2916, 3000, 3750, 4356, 5292, 6480, 6760, 8100, 8640, 9464, 9900, 10404, 10648, 11700, 12000, 12096, 13500, 14580, 14872, 15300, 15552, 15876, 16000, 16200, 16224, 17100, 17836, 18252, 19008, 19044, 20160, 20412, 20700, 21780, 22464, 22500

Offset: 1

Ant King, Jan 25 2007

nonn

Exponential aspiring numbers are those integers whose exponential aliquot sequences end in an e-perfect number, but that are not e-perfect numbers themselves.

			a(5) = 2916 because the fifth non-e-perfect number whose exponential aliquot sequence ends in an e-perfect number is 2916.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Peter Hagis, 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. A127656, A127657, A127659, A127660, A054979.

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]];Select[Range[25000],ExponentialPerfectNumberQ[Last[eTrajectory[ # ]]] && !ExponentialPerfectNumberQ[ # ]&]
f[p_, e_] := DivisorSum[e, p^# &]; s[0] = s[1] = 0; s[n_] := Times @@ f @@@ FactorInteger[n] - n; q[n_] := Module[{v = NestWhileList[s, n, UnsameQ, All]}, v[[-1]] != n && v[[-2]] == v[[-1]] > 0]; Select[Range[32000], q] (* Amiram Eldar, Mar 11 2023 *)

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A127660 Integers whose exponential aliquot sequences end in an exponential amicable pair.

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A127656 Lengths of the exponential aliquot sequences.

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A127657 Integers whose exponential aliquot sequences end in an e-perfect number.

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A127658 Exponential aspiring numbers.

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