A238382 Untouchable amicable numbers: amicable pairs which cannot be reached by any aliquot sequence starting from a number that does not belong to this pair.

356408, 399592, 643336, 652664, 5232010, 5799542, 9363584, 9437056, 10596368, 11199112, 15363832, 16517768, 31818952, 32205616, 34352624, 34860248, 46237730, 48641584, 48852176, 49215166, 52695376, 55349570, 56208368, 61319902, 91996816, 93259184

Offset

1,1

Comments

A pair of numbers x and y is called an untouchable amicable pair if x and y are amicable numbers (see A063990) and if x and y each have only one aliquot antecedent: the other number of their pair. In other words, they are amicable pairs which no aliquot sequence starting on a number that does not belong to this pair can reach.

The sequence lists the untouchable amicable numbers in increasing order. Note that the pairs x, y are not always adjacent to each other in the list.

Numbers that are the smaller number of their untouchable amicable pair are 356408, 643336, 5232010, 9363584, 10596368, 15363832, 31818952, 32205616, ... (subsequence of A002025).

The remaining numbers in the sequence are the larger number of their untouchable amicable pair: 399592, 652664, 5799542, 9437056, 11199112, 16517768, 34860248, 34352624, ... (subsequence of A002046).

We can call those pairs "untouchable amicable pairs", "isolated amicable pairs", or "isolated amicable cycles with two links" ... .

Links

Table of n, a(n) for n=1..26.

Jean-Luc Garambois, Aliquot sequences (in French, but with an English summary).

Jean-Luc Garambois, Python program

Crossrefs

Cf. A063990, A048138, A001065.

Keyword

nonn

Author

Jean Luc Garambois, Mar 03 2014

Status

approved

A363875 Numbers k such that there is no odd number whose aliquot sequence contains k.

2, 28, 52, 88, 96, 120, 124, 146, 162, 188, 206, 208, 210, 216, 238, 246, 248, 250, 262, 268, 276, 288, 290, 292, 298, 304, 306, 322, 324, 326, 336, 342, 362, 372, 388, 396, 406, 408, 412, 426, 428, 430, 438, 448, 452, 472, 474, 478, 486, 494, 498, 508, 516

Offset

1,1

Comments

k is in sequence iff k can never be reached when iterating the map x -> A001065(x) starting with any odd number m.

Assuming the stronger version of Goldbach conjecture, iff k is in the sequence, there are infinitely many odd numbers whose aliquot sequence contain k.

Supersequence of A005114 (except 5), A283152, A284147, A284156, A284187, ..., and untouchable perfect numbers (28, 137438691328, ...), untouchable amicable numbers (A238382), untouchable sociable numbers.

Links

Table of n, a(n) for n=1..53.

Andrew R. Booker, Finite connected components of the aliquot graph, arXiv:1610.07471 [math.NT], 2017.

Jean-Luc Garambois, Le graphe infini des suites aliquotes (in French).

Wikipedia, Aliquot sequence

Crossrefs

Cf. A001065, A005114, A238382, A283152, A284147, A284156, A284187.

Keyword

nonn,new

Author

Jinyuan Wang, Jun 25 2023

Status

approved