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Old 2021-09-10, 05:28   #12
sweety439
 
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Originally Posted by sweety439 View Post
We define "strong infinity-touchable number" as nonnegative integer n such that there exist infinitely many nonnegative integers whose Aliquot sequences reach n

e.g. all nonnegative integers <52 except 28 are infinity-touchable and strong infinity-touchable (assuming the stronger version of Goldbach conjecture), 28 and 356408 are infinity-touchable, but not strong infinity-touchable, 298 and 396 are neither infinity-touchable nor strong infinity-touchable.

All infinity-touchable numbers which are not strong infinity-touchable must be in an Aliquot cycle, i.e. must be perfect number, member of amicable pair, or member of sociable sequence.

A reference about infinity-touchable numbers which are not strong infinity-touchable: https://arxiv.org/pdf/1610.07471.pdf
All strong infinity-touchable numbers are infinity-touchable, since for all nonnegative integers n, there are only finitely many positive integers m such that s(m)=n (and the largest such m is at most (n-1)^2)

Besides, I computed all infinity-touchable numbers up to 351 (all such numbers except 28 are also strong infinity-touchable), they are the touchable numbers (nonnegative integers which are not untouchable are called touchable numbers) up to 351 except 208, 250, 298

Use the sense of https://arxiv.org/pdf/1610.07471.pdf to research (see https://oeis.org/A152454/a152454.txt), s^-1(208) = {268} and s^-1(268) = ∅, thus such sequence cannot exist, the same for 250 and 298

MODERATOR NOTE: Moved from Might miss small sociable number.

Last fiddled with by Dr Sardonicus on 2021-09-21 at 13:54
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Old 2022-01-13, 08:01   #13
sweety439
 
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I found that there are sequences about this in OEIS:

A283152 2-untouchable numbers: {208, 250, 362, 396, 412, 428, 438, 452, 478, 486, 494, 508, 672, 712, 716, 772, 844, 900, 906, 950, 1042, 1048, 1086, 1090, 1112, 1132, 1140, 1252, 1262, 1310, 1338, 1372, 1518, 1548, 1574, 1590, 1592, ...}

A284147 3-untouchable numbers: {388, 606, 696, 790, 918, 1264, 1330, 1344, 1350, 1468, 1480, 1496, 1634, 1688, 1800, 1938, 1966, 2006, 2026, 2202, 2220, 2318, 2402, 2456, 2538, 2780, 2830, 2916, 2962, 2966, 2998, ...}

A284156 4-untouchable numbers: {298, 1006, 1016, 1108, 1204, 1502, 1940, 2370, 2770, 3358, 3482, 3596, 3688, 3976, 4076, 4164, 4354, 5206, 5634, 5770, 6104, 6206, 6696, 6850, 7008, 7118, 7290, 7496, 7586, 7654, 7812, 7922, ...}

thus we can do a sequence: a(n) is the largest k such that n is in range of s^k, where s is A001065, and the infinity-touchable numbers are the numbers n such that a(n) = infinity (if we want to add this sequence to OEIS, then we can use -1 for infinity), this includes all odd numbers except 5 (assuming the stronger version of Goldbach conjecture (i.e. all even numbers >6 can be written as sum of two distinct primes), and all numbers in https://oeis.org/A078923 <= 351 except 208, 250, 298 (note: 28 is also included, since s^k(28) = 28 for all k, but if we require the aliquot sequence cannot contain repeating terms, then 28 will be excluded, and hence a(28) will be 0 instead of infinity, similarly, for the untouchable amicable number 356408, a(356408) will be 1 instead of infinity, since the only sequence is 399592, 356408, and the numbers n such that a(n) = infinity will be the strong infinity-touchable numbers instead of the infinity-touchable numbers)), the untouchable numbers n are the numbers n such that a(n) = 0, the 2-untouchable numbers are the numbers n such that a(n) = 1, the 3-untouchable numbers are the numbers n such that a(n) = 2, the 4-untouchable numbers are the numbers n such that a(n) = 3 (e.g. a(298) = 3, since the only sequence is 668, 508, 388, 298), etc.

Also, we can do another sequence for which b(n) = 1 for untouchable number n and hermit number (untouchable perfect number) n (like n = 28): b(n) is the number of numbers whose Aliquot sequence contain n, since the Aliquot sequence of n always contain n, b(n) >= 1 for all n, and the strong infinity-touchable numbers are the numbers n such that b(n) = infinity (if we want to add this sequence to OEIS, then we can use 0 for infinity, since the original b(n) is always >= 1 and cannot be 0), this includes all odd numbers except 5 (assuming the stronger version of Goldbach conjecture (i.e. all even numbers >6 can be written as sum of two distinct primes), and for the 2-untouchable numbers n, b(n) = A048138(n), and for the untouchable amicable numbers n like 356408, b(n) = 2

Last fiddled with by sweety439 on 2022-01-13 at 08:07
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Old 2022-01-13, 08:04   #14
sweety439
 
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Quote:
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problem: Is 28 the only such number?
No, see the graph in page 8 of https://arxiv.org/pdf/1610.07471.pdf, 137438691328 is also such number (untouchable perfect number, hermit number) like 28
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Old 2022-12-12, 13:33   #15
Andrew Usher
 
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That definition of 'strong infinity-untouchable' is probably the best for untouchability for the purpose of aliquot sequences. Numbers part of a 'finite connected component' meet that criterion in both directions, and have zero density. But what if we were to consider only even numbers, as that's more natural for the aliquot sequences we study? Is is still true that they have zero density among even numbers?

Similarly, to the original post of this thread I'd say that trying to continue a sequence backward makes sense only if done among even numbers (which should still be possible), as the task is essentially trivial for odd numbers by the method stated.

And since it's bothered me for a while, I'll say here that I think the 'stronger version of Goldbach's conjecture should now be just called Goldbach's conjecture. As it is the only version actually used anywhere else, it should have priority - and if you wanted to appeal to Goldbach's original statement, he also definitely considered 1 a prime and excluding that makes about as much difference to the strength of the statement. Besides, results about Goldbach partitions are more elegant when those of identical primes are excluded.
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Old 2022-12-14, 04:42   #16
sweety439
 
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Quote:
Originally Posted by Andrew Usher View Post
That definition of 'strong infinity-untouchable' is probably the best for untouchability for the purpose of aliquot sequences. Numbers part of a 'finite connected component' meet that criterion in both directions, and have zero density. But what if we were to consider only even numbers, as that's more natural for the aliquot sequences we study? Is is still true that they have zero density among even numbers?
Assuming the stronger version of Goldbach's conjecture, a number N is strong infinity-untouchable if and only if there is at least one odd number whose Aliquot sequence contains N, except N=5

I think that the strong infinity-untouchable numbers is an interesting sequence, but unfortunately there seems to be no sequence of the strong infinity-untouchable numbers in OEIS, the strong infinity-untouchable numbers contains all numbers < 352 not in https://oeis.org/A005114, except 28, 208, 250, 298
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Old 2022-12-14, 07:04   #17
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Quote:
Originally Posted by sweety439 View Post
...a number N is strong infinity-untouchable if and only if ...
Quote:
Originally Posted by https://dic.academic.ru/dic.nsf/dic_wingwords/1323/%D0%9A%D1%83%D0%BA%D1%83%D1%88%D0%BA%D0%B0
За что же, не боясь греха, кукушка хвалит петуха?
За то, что хвалит он кукушку
I.Krylov was mostly a creative interpreter of La Fontaine's fables. I leave this as an exercise to the reader to find Lafontaine's or Aesope's original fable. Perhaps, Ecclesiastes 7:5, but I don't know...
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