A127665 -id:A127665 - cp's OEIS frontend

A371422 Numbers whose aliquot-like sequence based on the largest aliquot divisor of the sum of divisors of n (A371418) terminates in a cycle of length 2.

12, 14, 15, 23, 29, 42, 44, 48, 54, 56, 60, 62, 65, 66, 69, 70, 72, 75, 76, 77, 78, 83, 84, 85, 86, 87, 88, 90, 91, 92, 94, 95, 99, 102, 107, 108, 110, 111, 112, 114, 115, 117, 118, 119, 120, 123, 124, 125, 128, 129, 131, 132, 134, 135, 136, 137, 139, 140, 142

Offset: 1

Amiram Eldar, Mar 23 2024

nonn

It is unknown whether 222 is a term of this sequence or not (see A371423).

			12 is a term because when we start with 12 and repeatedly apply the mapping x -> A371418(x), we get the sequence 12, 14, 12, 14, ...
76 is a term because when we start with 76 and repeatedly apply the mapping x -> A371418(x), we get the sequence 76, 70, 72, 65, 42, 48, 62, 48, 62, ...

Amiram Eldar, Table of n, a(n) for n = 1..108
Robert D. Carmichael, Empirical Results in the Theory of Numbers, The Mathematics Teacher, Vol. 14, No. 6 (1921), pp. 305-310; alternative link. See p. 309.

Cf. A371418, A371421, A371423.

Similar sequences: A127655, A127660, A127665.

r[n_] := n/FactorInteger[n][[1, 1]]; f[n_] := r[DivisorSigma[1, n]];
q[n_] := Module[{m = NestWhileList[f, n, UnsameQ, All][[-1]], k}, k = f[m]; k != m && f[k] == m]; Select[Range[221], q]

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 *)

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

cp's OEIS Frontend

A371422 Numbers whose aliquot-like sequence based on the largest aliquot divisor of the sum of divisors of n (A371418) terminates in a cycle of length 2.

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A127663 Infinitary aspiring numbers.

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A127664 Infinitary amicable numbers.

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