Relationship between A002093 & A036913

[Joe Crump]

Hi Sequence Fans...

	I came across an interesting sequence, whose increasing
values are the same as A036913, and the index into the sequence
for each new value is A002093 (almost). Perhaps some of you
can provide an analysis.

Here is the rule I followed to generate the sequence...

f(n) = Maximum m where we can find (x,y), 0<x<m, 0<y<m, satisfying 0 < xy
mod m <= n

Running a quick program generated the solutions...
======================================================
 m =   6, n= 1
 m =  12, n= 2
 m =  18, n= 3
 m =  30, n= 4..5
 m =  42, n= 6..7
 m =  60, n= 8..9
 m =  66, n=10..11
 m =  90, n=12..15
 m = 120, n=16..17
 m = 126, n=18..19
 m = 150, n=20..22
 m = 210, n=24..31
 m = 240, n=32..35
 m = 270, n=36..39
 m = 330, n=40..47
 m = 420, n=48..
======================================================

All of the m's match A036913, and the first n's where
a new m occurs matches the first 12 entries of A002093.
So it would appear at first f( A002093(n) ) = A036913(n+1).
However, after 12 entries A002093 seems to lose focus.

======================================================
For reference, here are A002093 & A036913...
======================================================
%I A036913
%S A036913
2,6,12,18,30,42,60,66,90,120,126,150,210,240,270,330,420,462,510,630,
%T A036913
660,690,840,870,1050,1260,1320,1470,1680,1890,2310,2730,2940,3150,
%U A036913
3570,3990,4620,4830,5460,5610,5670,6090,6930,7140,7350,8190,9240,9660
%N A036913 Increasing values of maximum inverse of phi(n) as n increases.
%O A036913 1,1%K A036913 nonn%A A036913 David Wilson (wilson at ctron.com)

%I A002093 M0553 N0200
%S A002093 1,2,3,4,6,8,10,12,16,18,20,24,30,36,42,48,60,72,84,90,96,108,120,
%T A002093
144,168,180,210,216,240,288,300,336,360,420,480,504,540,600,630,660,
%U A002093 720,840
%N A002093 Highly abundant numbers: sigma (n) > sigma (m) for all m < n.
%R A002093 TAMS 56 467 1944.
%D A002093 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical
Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and
various reprintings), p. 842.
%O A002093 1,2%A A002093 njas%K A002093 nonn,nice%Y A002093 Cf. A004394.
%E A002093 Better description 4/97.
======================================================

- Joe Crump (joecr at microsoft.com, mahoganyrock at msn.com)