Extending an aliquot sequence backwards

[arbooker]

The aliquot sequence with start value 461214 is the current longest known open ended sequence starting below 1e6 (it merges with the 4788 sequence around index 6000). Right now it's just shy of 19k iterations. Just for fun, I thought I would try to extend the sequence backwards using Goldbach so as to make a longer record.

It's easy to check that s(461214)=s(670097^2). Given an odd number n > 8, we can (conjecturally) write n=p+q+1=s(pq) for distinct primes p,q. Starting from n=670097^2, I found the smallest p such that q=n-1-p is a probable prime, replaced n by pq, and repeated for 1000 iterations. The sequence of p values is attached.

I then used Primo to certify the primality of the first 700 values of q, the largest of which has 2490 digits. It's getting very slow, but I might let it run up to 1000. I'm aware that none of this serves any real purpose, but if anyone would like to join me in this quest, feel free.

[garambois]

I did the same kind of exercise, see on this page (http://www.aliquotes.com/remonter_suite_envers.html) my go backwards of 3, 7, 11, 13, 17 on more than 2000 iterations, as well as the go backwards of 2005020 on more than 1000 iterations (I was talking about it here #7 : http://www.mersenneforum.org/showthread.php?t=18641)

[arbooker]

Originally Posted by garambois

I did the same kind of exercise

Yet more evidence that no idea is original. The only thing I can think to add is the primality certificates. It's reasonable to generate them with Primo for 1000 iterations or so.

Good idea putting the sequence on factordb; I thought it might be too large for that, but I guess not. Will factordb compute the sequence for me if I just enter the 3000-digit starting value? Also, I don't fancy uploading 1000 certificates by hand; if anyone has a script that they don't mind sharing, that would be a great help.

[sweety439]

The number 1578 merges to 56440, but it is not known whether the sequence of 56440 (currently at 166 digits) also merges to 56440, thus it is not known whether 56440 is member of sociable number cycle (but if so, then this cycle will contain 6000+ numbers ....)

[EdH]

Originally Posted by sweety439

The number 1578 merges to 56440, but it is not known whether the sequence of 56440 (currently at 166 digits) also merges to 56440, thus it is not known whether 56440 is member of sociable number cycle (but if so, then this cycle will contain 6000+ numbers ....)

factordb would pick up a cycle. Since all the factors of all the terms of a sequence exist within the database, it would not be shown as open ended. As to whether this eventually happens, only more factoring will tell. . .

[sweety439]

so the sequences A003416 and A122726 might miss numbers (i.e. it is unknown whether there are members of sociable number cycle <= 1799281330 other than numbers listed in A003416), however, the smallest number whose aliquot sequence has not yet been fully determined (276) cannot be member of sociable number cycle, since it is an untouchable number (A005114), however, the number 56440 might be.

[sweety439]

The smallest possible missing sociable number is 564, i.e. the smallest number which it is unknown whether this number is in a sociable cycle or not is 564 (but of course, if 564 is in a sociable cycle, then this cycle will contain >3486 numbers and contain the number 316969 = 563^2), since in the list https://oeis.org/A131884 (numbers whose aliquot sequence has not yet been fully determined):

276 is untouchable number
306 is untouchable number
396 is not untouchable, but if s(n) = 396, then n is either 276 or 306 (276 and 306 are the only two values of n such that s(n) = 396), but both 276 and 396 are untouchable numbers, thus 396 is not possible in a sociable cycle
552 is untouchable number
564 is not untouchable, the only value of n such that s(n) = 564 is 316969 = 563^2, and given an odd number n > 7, we can (conjecturally) write n=p+q+1=s(p*q) for distinct primes p, q, and we can find many pairs of primes (p, q) such that p+q=316968, and all p*q of these (p, q) merge with the 564 sequence, the such (p, q) pair and p*q values are attached, there are 3117 such values, the smallest such value is 2218727, and the largest such value is 25117177031

On this research, we can extend the Aliquot sequences backwards and define "infinity-touchable number" as nonnegative integer n such that there exists such infinite sequence of nonnegative integers: s(a_1) = n, s(a_2) = a_1, s(a_3) = a_2, s(a_4) = a_3, ..., s(a_(i+1)) = a_i for all integer i>=1, clearly, if a stronger version of Goldbach conjecture (i.e. all even numbers >6 can be written as sum of two distinct primes) is true, then every positive odd number (except 5) are infinity-touchable, and all numbers which are of the form p+1 or p+3 with prime p (except 5) are infinity-touchable, however, 396 is not infinity-touchable since if s(a_1) = 396, then a_1 can only be 276 or 306, and for these two values of a_1, no values of a_2 exists to make s(a_2) = a_1, but 564 is infinity-touchable if this stronger version of Goldbach conjecture is true.

Two interesting ones: For n = 28, the only such infinite sequence is 28, 28, 28, 28, ..., since 28 is the only number n such that s(n) = 28 (this is not true for all perfect numbers), and for n = 220 and 284, the only such infinite sequence is 220, 284, 220, 284, ..., since 284 is the only number n such that s(n) = 220, and 220 and 562 are the only two numbers n such that s(n) = 284, but 562 is untouchable number (this is not true for all amicable numbers)

Question: If this stronger version of Goldbach conjecture is true, what is the sequence of the infinity-touchable numbers?

MODERATOR NOTE: Moved from Might miss small sociable number.

[sweety439]

Found a reference!!! https://math.dartmouth.edu/~carlp///sociabletalk2.pdf

Also see related sequence of Aliquot sequences

MODERATOR NOTE: Moved from Might miss small sociable number.

[sweety439]

Originally Posted by sweety439

Question: If this stronger version of Goldbach conjecture is true, what is the sequence of the infinity-touchable numbers?

This is the PARI/GP program for this question: (a(n) is the row n for A152454)

a(n)=v=[];for(k=1,n^2,if(sigma(k)-k==n,v=concat(v,k)));v
is(n)=if(n==0 || sigma(n)==2*n || (n%2 && n != 5),1,for(k=1,length(a(n)),if(is(a(n)[k]),return(1)));0)

Type is(n) to test whether a nonnegative integer n is infinity-touchable number, assuming the stronger version of Goldbach conjecture (i.e. all even numbers >6 can be written as sum of two distinct primes).

MODERATOR NOTE: Moved from Might miss small sociable number.

[sweety439]

Originally Posted by sweety439

Two interesting ones: For n = 28, the only such infinite sequence is 28, 28, 28, 28, ..., since 28 is the only number n such that s(n) = 28 (this is not true for all perfect numbers), and for n = 220 and 284, the only such infinite sequence is 220, 284, 220, 284, ..., since 284 is the only number n such that s(n) = 220, and 220 and 562 are the only two numbers n such that s(n) = 284, but 562 is untouchable number (this is not true for all amicable numbers)

No, there are three n such that s(n) = 284: 220, 562, 80089=283^2, and if the stronger version of Goldbach conjecture is true, then there are infinitely many numbers whose Aliquot sequence reach the (220, 284) pair.

However, there are amicable pairs which cannot be reached by any aliquot sequence starting from a number that does not belong to this pair, they are listed in the OEIS sequence A238382, the untouchable amicable numbers, the smallest such amicable pair is (356408, 399592). The number 28 is a perfect number which cannot be reached by any aliquot sequence starting from other number, such numbers can be called untouchable perfect numbers, or hermit numbers (see https://oeis.org/A057709), problem: Is 28 the only such number?

MODERATOR NOTE: Moved from Might miss small sociable number.

[sweety439]

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 not 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

MODERATOR NOTE: Moved from Might miss small sociable number.