[seqfan] Re: min k: tau^n(k) = 2

[hv at crypt.org]

Sorry, I'm not sure which part you are finding cryptic.

b(k) asks how many times you must iterate k -> tau(k) before you
reach 2; a(n) finds records in b(k).

I have found the sequences A036459, A251483 which turn out to be
a minor variant of the sequences I was considering (they replace
tau(m) = 2 with tau(m) = m), so I've proposed adding a reference
to the latter in A009287 and in A036460.

I have not attempted to verify the correctness of either A009287
or A251483, and have no particular opinion on whether the former
should include 2. I do know that A009287 is not the sequence I was
considering. (When I have time I may check A251483 and see if I
can extend it.)

Hugo

"M. F. Hasler" <seqfan at hasler.fr> wrote:
:- Maximilian
:
:
:On Tue, Nov 15, 2022 <hv at crypt.org> wrote:
:
:> Last night, though, I considered "min k: tau^n(k) = 2", and I think
:> A036460 would be a correct rendition of this sequence with offset 0.
:>
:> For the avoidance of doubt, of course the subject is wrong; it should
:> rather be:
:>   let b(k) = min m: tau^m(k) = 2, m >= 0
:>   then a(n) = min k: b(k) = n
:>
:
:This looks extremely cryptic to me.
:Why not "restore" A009287(0) = 2
:and simply amend the definition to "... least k > a(n) with a(n) divisors"
:(or change the definition of A036460 to this)?
:
:It does not look as if that sequence could be not strictly increasing after
:the term "3"...(?)
:[EDIT : I see that this suggestion was already made by  Hal M. Switkay
:<https://oeis.org/wiki/User:Hal_M._Switkay>, Jul 03 2022]
:
:But more important: are we even sure that the terms are correct?!?
:
:A009287 : a(1) = 3; thereafter a(n+1) = least k with a(n) divisors.
:DATA : 3, 4, 6, 12, 60, 5040, 293318625600,
:670059168204585168371476438927421112933837297640990904154667968000000000000
:COMMENT :
:The calculation of a(7) and a(8) is based upon the method in A037019
:(which, apparently, is the method previously used by the authors of
:A009287). So a(7) and a(8) are correct unless a(6) = 5040 or a(7) =
:293318625600 are "exceptional" as described in A037019.
:
:
:This "a(7) is exceptional" means EXACTLY the same as "a(8) is incorrect",
:namely, it means that the least number with 293318625600 divisors is less,
:thus NOT equal to
:A037019 ( 293318625600 ) =
:670059168204585168371476438927421112933837297640990904154667968000000000000
:which is the conjectured (?! wild guess !?) a(8).
:
:As written in A037019:
:This is an *easy way* to produce a number with exactly n divisors...
:
:but the (so-called exceptional) counter-examples oeis.org/A072066
:are extremely frequent: about one in 30 numbers : ~500 below 15'000 and
:~1000 below 30'000.
:
:I would not bet anything on this wild guess for a(8),
:if necessary, I would even bet against, for the fun.
:(The comment by Jianing Song <https://oeis.org/wiki/User:Jianing_Song>, Jul
:15 2021, seems to go in the same sense,
:exhibiting a *much* smaller number having *more* divisors.)
:
:- Maximilian
:
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