cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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

Original entry on oeis.org

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

Views

Author

Ant King, Jan 25 2007

Keywords

Comments

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.

Examples

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

Links

Crossrefs

Cf. A127656, A127657, A127658.
Subsequences: A127659, A126165, A126166.

Programs

  • Mathematica
    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 *)

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

Views

Author

Ant King, Jan 25 2007

Keywords

Examples

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

Links

Crossrefs

Cf. A127656, A127658, A127659, A127660, A054979.

Programs

  • Mathematica
    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

Views

Author

Ant King, Jan 25 2007

Keywords

Comments

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

Examples

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

Links

Crossrefs

Cf. A127656, A127657, A127659, A127660, A054979.

Programs

  • Mathematica
    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 *)

A127659 Exponential amicable numbers.

Original entry on oeis.org

90972, 100548, 454860, 502740, 937692, 968436, 1000692, 1106028, 1182636, 1307124, 1546524, 1709316, 2092356, 2312604, 2638188, 2820132, 2915892, 3116988, 3365964, 3720276, 3729852, 3911796, 4122468, 4275684, 4323564, 4548600, 4688460, 4725756, 4821516, 4842180
Offset: 1

Views

Author

Ant King, Jan 25 2007

Keywords

Comments

Union of A126165 and A126166. The first 10 terms of this sequence are the same as the first 10 terms of A127660.

Examples

			a(5)=937692 because the fifth non-e-perfect integer that satisfies A126164(A126164(n))=n is 937692.

References

  • Hagis, Peter Jr.; Some Results Concerning Exponential Divisors, Internat. J. Math. & Math. Sci., Vol. 11, No. 2, (1988), pp. 343-350.

Links

Crossrefs

Cf. A126164, A126165, A126166, A127656, A127657, A127658, A127660, A051377.

Programs

  • Mathematica
    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[ # ] &]
fun[p_, e_] := DivisorSum[e, p^# &]; esigma[1] = 1; esigma[n_] := Times @@ fun @@@ FactorInteger[n]; s = {}; Do[m = esigma[n] - n; If[m != n && esigma[m] - m == n, AppendTo[s, n]], {n, 1, 10^6}]; s (* Amiram Eldar, May 09 2019 *)

Formula

Non-e-perfect numbers for which A126164(A126164(n))=n.

Extensions

Link corrected by Andrew Lelechenko, Dec 04 2011
More terms from Amiram Eldar, May 09 2019
Showing 1-4 of 4 results.

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