![]() |
![]() |
#12 | |
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
71428 Posts |
![]() Quote:
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 |
|
![]() |
![]() |
![]() |
#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 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 |
![]() |
![]() |
![]() |
#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
|
![]() |
![]() |
![]() |
#15 |
Dec 2022
3218 Posts |
![]()
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. |
![]() |
![]() |
![]() |
#16 | |
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
2·7·263 Posts |
![]() Quote:
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 |
|
![]() |
![]() |
![]() |
#17 | ||
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
32·1,117 Posts |
![]() Quote:
Quote:
|
||
![]() |
![]() |
![]() |
Thread Tools | |
![]() |
||||
Thread | Thread Starter | Forum | Replies | Last Post |
Aliquot sequence reservations | schickel | Aliquot Sequences | 3744 | 2023-01-30 08:31 |
Useful aliquot-sequence links | 10metreh | Aliquot Sequences | 4 | 2021-10-28 22:17 |
Another Aliquot Sequence site | schickel | Aliquot Sequences | 67 | 2012-01-20 17:53 |
Aliquot sequence worker for factordb | yoyo | FactorDB | 6 | 2012-01-12 20:58 |
Aliquot sequence convergence question | philmoore | Math | 3 | 2009-03-20 19:04 |