[IntSeq] Friends of R and friends of S

[Rainer Rosenthal]

Gottfried Helms <helms at uni-kassel.de> wrote

> > Have fun.
> >
> The R-sequences form an interesting "divisors-tree".
> [ tree with surprising resemblance to 3x+1 trees :-))) ]

I'm glad you pointed that out: I already thought about the
number theoretic consequences, hoping for interesting remarks.
Thanks for this first one of many more to come ...

> But how do you show, that more than this one sequence
> (except recursions, which excludes periodic sequences)
> are of infinite length?

Humm, err, ... good question. But maybe the comments in
the OEIS will help. As a SeqFan you already know these
lists, but I am glad to present them here in a.m.r.:

Chains of R-friends:
R_2 = 0,1,2,3,4,5,6,...      = "the natural numbers"
R_3 = 0,1,3,8,21,55,144,...  = A001906
R_4 = 0,1,4,15,56,209,...    = A001353
R_5 = 0,1,5,24,115,551,...   = A004254
R_6 = 0,1,6,35,204,1189,...  = A001109
R_7 = A004187  R_8 = A001090  R_9 = A018913
R_10 = A004189  R_11 = A004190  R_12 = A004191
R_13 = A078362  R_14 = A007655  R_15 = A078364
R_16 = A077412  R_17 = A078366  R_18 = A049660
R_19 = A078368  R_20 = A075843  R_21 = A092499(*)
R_22 = A077421

Chains of S-Friends:
S_4 = 0,1,4,9,16,25,36,49,...  = A000290
S_5 = 0,1,5,16,45,121,320,...  = A004146
S_6 = 0,1,6,25,96,361,1350,...  = A092184(*)
S_7 = 0,1,7,36,175,841,4032,...  = A054493
S_8 = 0,1,8,49,288,1681,9800,...   = A001108
S_9 = 0,1,9,64,441,3025,20736,...  = A049684
S_20 = 0,1,20,361,6480,116281,...  = A049683
S_36 = 0,1,36,1225,41616,1413721,... = A001110(x)
S_49 = 0,1,49,2304,108241, ...     = A049682
S_144 = 0,1,12^2,143^2,1704^2,...  = A004191^2

(*) first sequence of this type not already in the OEIS.
(x) This started these observations

Playing with all these diamonds and pearls as if they
were marbles, I thought it might be nice to define:

    "A is R-fond of B" if A divides B^2-1

because then we can say:

    A and B are R-friends iff they
    are mutually R-fond of each other

Likewise for S-friends:

    "A is S-fond of B" if A divides (B-1)^2

because then we can say:

    A and B are S-friends iff they
    are mutually S-fond of each other

One interesting question, which I didn't investigate until
now, but which seems correlated with your inifinity-question:
Are there isolated friends? I.e. A and B friends without any
other friend for B?

Best regards from a happy player
Rainer Rosenthal
r.rosenthal at web.de