R. J. Cano (in private email communications with me) provided answers to both of my questions.
In the first email he expressed his intuitive prediction that in the base 10 could be found more pairs (he generously named them PovolotskyPairs) of such permutations (with no duplicates), which yield the vary same ratio 109739369/13717421 = 8.0000000729... .
In the same email R. J. Cano expressed his prediction that permutation pairs with similar properties (but not with same as in base 10 ratio) also could be found for other bases.
In his second email R. J. Cano informed me that after spending one day he (using 2 PARI programs, which he wrote) produced the following results for such pairs of permutations found for all bases from 2 to 10 (outputted in decimal, that is in base 10 format) :
? assumedBase=2;
? reportPovolotskyPairs()
Loading permutations...
Performing calculations...
Reporting the pairs:
1) {1,1};
2) {2,2};
Ratio 1
In base 2 notation 1) {01, 01}; 2) {10, 10};
Sorted in increasing order (and divided by d=1) Multiplication factors (to get base 2
PovolotskyPairs from the ratio):
k(m) = {1, 2}
?
? assumedBase=3;
? reportPovolotskyPairs()
Loading permutations...
Performing calculations...
Reporting the pairs:
1) {7,5};
2) {21,15};
Ratio 7/5 = 1.4
In base 3 notation 1) {021,012}; 2) {210,120};
Sorted in increasing order (and divided by d=1) Multiplication factors (to get base 3
PovolotskyPairs from the ratio):
k(m) = {1, 3}
?
? assumedBase=4;
? reportPovolotskyPairs()
Loading permutations...
Performing calculations...
Reporting the pairs:
1) {114,54};
2) {57,27};
3) {228,108};
Ratio 19/9 = 2.111...
In base 4 notation 1) {1302, 0312}; 2) {0321,0123}; 3) {3210,1230};
Sorted in increasing order (and divided by d=3) Multiplication factors (to get base 4
PovolotskyPairs from the ratio):
k(m) = {1, 2, 4}
?
? assumedBase=5;
? reportPovolotskyPairs()
Loading permutations...
Performing calculations...
Reporting the pairs:
1) {1758,582};
2) {586,194};
3) {2930,970};
Ratio 293/97=3.0206...
In base 5 notation 1) {24013, 04312}; 2) {04321,01234}; 3) {43210,12340)
Sorted in increasing order (and divided by d=2) Multiplication factors (to get base 5
PovolotskyPairs from the ratio):
k(m) = {1, 3, 5}
which are described by formula
k(m) = 2*m -1 for m=1,2,3
?
? assumedBase=6;
? reportPovolotskyPairs()
Loading permutations...
Performing calculations...
Reporting the pairs:
1) {29860,7460};
2) {22395,5595};
3) {14930,3730};
4) {7465,1865};
5) {44790,11190};
Ratio 1493/373=4.00268...
In base 6 notation 1) {350124,054312}; 2) {251403, 041523}; 3)
{153042,025134}; 4) {054321,012345}; 5) {543210,123450};
Sorted in increasing order (and divided by d=5) Multiplication factors (to get base 6
PovolotskyPairs from the ratio):
k(m) = {1, 2, 3, 4, 6}
?
? assumedBase=7;
? reportPovolotskyPairs()
Loading permutations...
Performing calculations...
Reporting the pairs:
1) {571905,114375};
2) {114381,22875};
3) {800667,160125};
Ratio 38127/7625=5.000262...
In base 7 notation 1) {4601235,0654312}; 2) {0654321,0123456}; 3)
{6543210,1234560}
Sorted in increasing order (and divided by d=3) Multiplication factors (to get base 7
PovolotskyPairs from the ratio):
k(m) = {1, 5, 7)
? assumedBase=8;
? reportPovolotskyPairs()
Loading permutations...
Performing calculations...
Reporting the pairs:
1) {12326118,2054346};
2) {10271765,1711955};
3) {8217412,1369564};
4) {6163059,1027173};
5) {4108706,684782};
6) {2054353,342391};
7) {16434824,2739128};
Ratio 293479/48913 = 6.000020...
In base 8 notation 1) {57012346, 07654312}; 2) {47136025, 06417523};
3){37261504,05162734};4) {27405163,03726145}; 5) {17530642,02471356};
6) {07654321,01234567}; 7) {76543210,12345670};
Sorted in increasing order (and divided by d=7) Multiplication factors (to get base 8
PovolotskyPairs from the ratio):
k(m) = {1, 2, 3, 4, 5, 6, 8}
? assumedBase=9;
? reportPovolotskyPairs()
Loading permutations...
Performing calculations...
Reporting the pairs:
1) {296618812,42374108};
2) {211870580,30267220};
3) {127122348,18160332};
4) {42374116,6053444};
5) {381367044,54480996};
Ratio = 10593529/1513361=7.00000132...
In base 9 notation 1) {680123457,087654312}; 2) {482603715,
062851734}; 3) {285174063,037148256}; 4) {087654321,012345678}; 5)
{876543210,123456780};
Sorted in increasing order (and divided by d=4) Multiplication factors (to get base 9
PovolotskyPairs from the ratio):
k(m) = {1, 3, 5, 7, 9}
which are described by formula
k(m) = 2*m - 1 for m=1,2,3,4,5
? assumedBase=10;
? reportPovolotskyPairs()
Loading permutations...
Performing calculations...
Reporting the pairs:
1) {7901234568,0987654312};
2) {6913580247,0864197523};
3) {4938271605,0617283945};
4) {3950617284,0493827156};
5) {1975308642,0246913578};
6) {0987654321,0123456789};
7) {9876543210,1234567890};
Ratio = 109739369/13717421 = 8.0000000729...
Sorted in increasing order (and divided by d=9) Multiplication factors (to get base 10
PovolotskyPairs from the ratio):
k(m) = { 2, 4, 5, 7, 8, 10}
which are described by formula
k(m) = (3 + (-1)^m + 6*m)/4 for m=1,2,3,4,5,6
Analyzing above R. J. Cano results - it is interesting to observe seven facts:
1) The number of "PovolotskyPairs" is:
{2,2,3,3,5,3,7,5,7,...} for n>=1 where n=r-1 where r is the base radix
Judging by above sequence it appears that the number of "PovolotskyPairs" is related to phi, which is the Euler totient function - see A039649, A039650, A214288 in the OEIS.
2) The ratio increases in its value as base grows
3) Starting from base 3 and going to base 10, the ratio is getting closer to exact integer value
4) The sequence of gcd dividers d for covered by calculations range of bases
r = {2, 3, 4, 5, 6, 7, 8, 9, 10}
is d = {1, 1, 3, 2, 5, 3, 7, 4, 9}
which is described by formula
d(r) = -((-3 + (-1)^(r-1))*(r-1))/4 for r >= 2
which gives d(r) sequence continuation as:
d = {1, 1, 3, 2, 5, 3, 7, 4, 9, 5, 11, 6, 13, 7, 15, 8, 17, 9, 19, 10, 21, 11, 23, 12, ...}
which is A026741 in OEIS
5) I sent (over email) to OEIS superseeker the sequence consisting of numerators of obtained ratios and received following reply:
Report on [ 1,7,19,293,1493,38127,293479,10593529,109739369]:
SUCCESS
Transformation T006 gave a match with the following sequences (up to a limit of 10):
%I A051847
%S A051847 1,19,1493,293479,109739369,66987982331,60693710471869,
%T A051847 76519827268721103,128138108936443028945,275176672984400058317539,
%U A051847 737345594135016860806925221,2411620538399461719230688945719
%N A051847 Bisection of A051846, divided by the term position.
%F A051847 a(n) = A051846(2n-1)/(2n-1)
%A A051847 Antti Karttunen Dec 13 1999
List of transformations used:
T006 elements of odd index in the sequence
More detailed analysis revealed that the sequence of ratio's numerators is described as:
a(n) = -4*((n-1)*(n+1)^(n+1)+1)/(((-1)^n-3)*n^3) for n=1 ... infinity
it could also be expressed as:
a(n) = -4*A051846(n)/((-3 + (-1)^n)*n) for n=1 ... infinity
First 30 terms are:
{1, 7, 19, 293, 1493, 38127, 293479, 10593529, 109739369, 5135610071, 66987982331, 3856048810781, 60693710471869, 4149140360751583, 76519827268721103, 6058888636862818097, 128138108936443028945, 11533996620790579909159, 275176672984400058317539, 27752288337333416976560341, 737345594135016860806925221, 82360974454696873163308732367, 2411620538399461719230688945719, 295545481092344912381290837570473, 9455799675198193629383691706982713, 1261454506836133847144542645138683127, 43783142861585517778127873524741134539, 6316283357812746048940357306645174509389, 236375074773470498995448767887162245274509, 36667114799331395762941381321682972438910911}
This sequence is submitted into OEIS as A221740
6) Also I sent (over email) to OEIS superseeker the sequence consisting of denominators of obtained ratios and received following reply:
Report on [ 1,5,9,97,373,7625,48913,1513361,13717421]:
SUCCESS
Transformation T006 gave a match with the following sequences :
%I A051848
%S A051848 1,9,373,48913,13717421,6698798233,5057809205989,5465701947765793,
%T A051848 8008631808527689309,15287592943577781017641,
%U A051848 36867279706750843040346261,109619115381793714510485861169
%N A051848 Bisection of A023811, divided by the term position.
%F A051848 a(n) = A023811[2n-1]/(2n-1)
%Y A051848 Cf. A051847.
%A A051848 Antti Karttunen Dec 13 1999
More detailed analysis revealed that the sequence of ratio's denominators is described as:
a(n) = -4*(((n+1)^(n+1)-(n+1))/((n+1)-1)^2-1)/((-3+(-1)^n)*n) for n=1 ... infinity
it could also be expressed as:
a(n) = -4*A023811(n+1)/((-3 + (-1)^n)*n) for n=1 ... infinity
First 30 terms are:
{1, 5, 9, 97, 373, 7625, 48913, 1513361, 13717421, 570623341, 6698798233, 350549891889, 5057809205989, 319164643134737, 5465701947765793, 403925909124187873, 8008631808527689309, 678470389458269406421, 15287592943577781017641, 1460646754596495630345281, 36867279706750843040346261, 3921951164509374912538511065, 109619115381793714510485861169, 12849803525754126625273514676977, 393991653133258067890987154457613, 50458180273445353885781705805547325, 1683967033137904529927995135566966713, 233936420659731335145939159505376833681, 8441966956195374964123170281684365902661, 1264383268942461922860047631782171463410721}
This sequence is submitted into OEIS as A221741
7) As a result first 30 ratios are:
{1, 7/5, 19/9, 293/97, 1493/373, 38127/7625, 293479/48913, 10593529/1513361, 109739369/13717421, 5135610071/570623341, 66987982331/6698798233, 3856048810781/350549891889, 60693710471869/5057809205989, 4149140360751583/319164643134737, 76519827268721103/5465701947765793, 6058888636862818097/403925909124187873, 128138108936443028945/8008631808527689309, 11533996620790579909159/678470389458269406421, 275176672984400058317539/15287592943577781017641, 27752288337333416976560341/1460646754596495630345281, 737345594135016860806925221/36867279706750843040346261, 82360974454696873163308732367/3921951164509374912538511065, 2411620538399461719230688945719/109619115381793714510485861169, 295545481092344912381290837570473/12849803525754126625273514676977, 9455799675198193629383691706982713/393991653133258067890987154457613, 1261454506836133847144542645138683127/50458180273445353885781705805547325, 43783142861585517778127873524741134539/1683967033137904529927995135566966713, 6316283357812746048940357306645174509389/233936420659731335145939159505376833681, 236375074773470498995448767887162245274509/8441966956195374964123170281684365902661, 36667114799331395762941381321682972438910911/1264383268942461922860047631782171463410721}
The 30th ratio in decimals is:
29.00000000000000000000000000000000000000000158...
What is left undone is analytical proof of my following conjecture:
Every complete set of all possible digits permutations (with no duplicates) in any numerical base (radix) r contains some prime number of permutations pairs (PovolotskyPairs), which have the same specific and unique ratio value defined as
(n^2 (n+1)^n-(n+1)^n+1) / (-n^2+n (n+1)^n+(n+1)^n-n-1)
for n=1 ... infinity where n = r - 1
