20210910, 05:28  #12  
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
7142_{8} Posts 
Quote:
Besides, I computed all infinitytouchable numbers up to 351 (all such numbers except 28 are also strong infinitytouchable), 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 20210921 at 13:54 

20220113, 08:01  #13 
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
2·7·263 Posts 
I found that there are sequences about this in OEIS:
A283152 2untouchable 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 3untouchable 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 4untouchable 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 infinitytouchable 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 infinitytouchable numbers instead of the infinitytouchable numbers)), the untouchable numbers n are the numbers n such that a(n) = 0, the 2untouchable numbers are the numbers n such that a(n) = 1, the 3untouchable numbers are the numbers n such that a(n) = 2, the 4untouchable 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 infinitytouchable 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 2untouchable numbers n, b(n) = A048138(n), and for the untouchable amicable numbers n like 356408, b(n) = 2 Last fiddled with by sweety439 on 20220113 at 08:07 
20220113, 08:04  #14 
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
2×7×263 Posts 
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

20221212, 13:33  #15 
Dec 2022
321_{8} Posts 
That definition of 'strong infinityuntouchable' 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. 
20221214, 04:42  #16  
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
2·7·263 Posts 
Quote:
I think that the strong infinityuntouchable numbers is an interesting sequence, but unfortunately there seems to be no sequence of the strong infinityuntouchable numbers in OEIS, the strong infinityuntouchable numbers contains all numbers < 352 not in https://oeis.org/A005114, except 28, 208, 250, 298 

20221214, 07:04  #17  
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
3^{2}·1,117 Posts 
Quote:
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