Talk:Rare numbers
comments concerning interesting observations from an webpage
(The author's webpage, the last URL reference from this task's preamble, re-shown below:)
(a URL reference):
- author's website: rare numbers by Shyam Sunder Gupta. (lots of hints and some observations).
I was considering adding checks (to the REXX program) to assert that for:
- when the number of digits in a rare number is even, the sum must be divisible by 11, and
- when the number of digits in a rare number is odd, the difference must be divisible by 9.
In fact, all the other (previous) checks (in the REXX program) have already filtered out the two (above) wrong cases, so the above two interesting observations are never observed to be false, so the checks are (at that point) in fact, redundant.
In fact, the webpage section contains a errors, the difference must be divisible by 9, not 11 as it states in the text.
Also, the mention of A2 and B2 having to be divisible by some number seems to be also wrong. I'm attempting to contact the author via e-mail. -- Gerard Schildberger (talk) 20:59, 8 September 2019 (UTC)
the 1st REXX version
This is the 1st REXX version, before all the optimizations were added: <lang rexx>/*REXX program to calculate and display an specified amount of rare numbers. */ numeric digits 20; w= digits() + digits() % 3 /*ensure enough decimal digs for calcs.*/ parse arg many start . /*obtain optional argument from the CL.*/ if many== | many=="," then many= 3 /*Not specified? Then use the default.*/
- = 0 /*the number of rare numbers (so far)*/
do n=10 /*N=10, 'cause 1 dig #s are palindromic*/ r= reverse(n) /*obtain the reverse of the number N. */ if r>n then iterate /*Difference will be negative? Skip it*/ if n==r then iterate /*Palindromic? Then it can't be rare.*/ s= n+r /*obtain the sum of N and R. */ d= n-r /* " " difference " " " " */ if iSqrt(s)**2 \== s then iterate /*Not a perfect square? Then skip it. */ if iSqrt(d)**2 \== d then iterate /* " " " " " " " */ #= # + 1 /*bump the counter of rare numbers. */ say right( th(#), length(#) + 9) ' rare number is: ' right( commas(n), w) if #>=many then leave /* [↑] W: the width of # with commas.*/ end /*n*/
exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ commas: parse arg _; do jc=length(_)-3 to 1 by -3; _=insert(',', _, jc); end; return _ th: parse arg th;return th||word('th st nd rd',1+(th//10)*(th//100%10\==1)*(th//10<4)) /*──────────────────────────────────────────────────────────────────────────────────────*/ iSqrt: parse arg x; $= 0; q= 1; do while q<=x; q= q*4
end /*while q<=x*/
do while q>1; q= q % 4; _= x-$-q; $= $ % 2
if _>=0 then do; x= _; $= $ + q
end
end /*while q>1*/; return $</lang>
Pretty simple, but slow as molasses in January.
Not ready for prime time.
the 2nd REXX version
This is the 2nd REXX version, after all of the hints (properties of rare numbers) within Shyam Sunder Gupta's webpage have been incorporated in this REXX program. <lang rexx>/*REXX program to calculate and display an specified amount of rare numbers. */ numeric digits 20; w= digits() + digits() % 3 /*ensure enough decimal digs for calcs.*/ parse arg many start . /*obtain optional argument from the CL.*/ if many== | many=="," then many= 5 /*Not specified? Then use the default.*/
@dr.=0; @dr.2= 1; @dr.5=1 ; @dr.8= 1; @dr.9= 1 /*rare # must have these digital roots.*/ @ps.=0; @ps.2= 1; @ps.3= 1; @ps.7= 1; @ps.8= 1 /*perfect squares must end in these.*/ @end.=0; @end.1=1; @end.4=1; @end.6=1; @end.9=1 /*rare # must not end in these digits.*/ @dif.=0; @dif.2=1; @dif.3=1; @dif.7=1; @dif.8=1; @dif.9=1 /* A─Q mustn't be these digs.*/ @noq.=0; @noq.0=1; @noq.1=1; @noq.4=1; @noq.5=1; @noq.6=1; @noq.9=1 /*A=8, Q mustn't be*/ @149.=0; @149.1=1; @149.4=1; @149.9=1 /*values for Z that need a even Y. */
- = 0 /*the number of rare numbers (so far)*/
@n05.=0; do i= 1 to 9; if i==0 | i==5 then iterate; @n05.i= 1; end /*¬1 ¬5*/ @eve.=0; do i=-8 by 2 to 8; @eve.i=1; end /*define even " some are negative.*/ @odd.=0; do i=-9 by 2 to 9; @odd.i=1; end /* " odd " " " " */
/*N=10, 'cause 1 dig #s are palindromic*/ do n=10; parse var n a 2 b 3 -2 p +1 q /*get 1st\2nd\penultimate\last digits. */ if @end.q then iterate /*rare numbers can't end in: 1 4 6 or 9*/ if q==3 then iterate
select /*weed some integers based on 1st digit*/
when a==q then do
if a==2|a==8 then nop /*if A = Q, then A must be 2 or 8. */
else iterate /*A not two or eight? Then skip.*/
if b\==p then iterate /*B not equal to P? Then skip.*/
end
when a==2 then do; if q\==2 then iterate /*A = 2? Then Q must also be 2. */
if b\==p then iterate /*" " " Then B must equal P. */
end
when a==4 then do
if q\==0 then iterate /*if Q not equal to zero, then skip it.*/
_= b - p /*calculate difference between B and P.*/
if @eve._ then iterate /*Positive not even? Then skip it.*/
end
when a==6 then do
if @n05.q then iterate /*Q not a zero or five? Then skip it.*/
_= b - p /*calculate difference between B and P.*/
if @eve._ then iterate
end
when a==8 then do
if @noq.q then iterate /*Q isn't one of 2, 3, 7, 8? Skip it.*/
select
when q==2 then if b+p\==9 then iterate
when q==3 then do; if b>p then if b-p\== 7 then iterate
else if b
1 then if b+p\==11 then iterate
else if b==0 then if b+p\== 1 then iterate
end
when q==8 then if b\==p then iterate
otherwise nop
end /*select*/
end /* [↓] A is an odd digit. */
otherwise n= n + 10**(length(n) - 1) - 1 /*bump N so next N starts with even dig*/
iterate /*Now, go and use the next value of N.*/
end /*select*/
_= a - q; if @dif._ then iterate /*diff of A─Q must be: 0, 1, 4, 5, or 6*/
r= reverse(n) /*obtain the reverse of the number N. */
if r>n then iterate /*Difference will be negative? Skip it*/
if n==r then iterate /*Palindromic? Then it can't be rare.*/
d= n-r; parse var d -2 y +1 z /*obtain the last 2 digs of difference.*/
if @ps.z then iterate /*Not 0, 1, 4, 5, 6, 9? Not perfect sq.*/
select
when z==0 then if y\==0 then iterate /*Does Z = 0? Then Y must be zero. */
when z==5 then if y\==2 then iterate /*Does Z = 5? Then Y must be two. */
when z==6 then if y//2==0 then iterate /*Does Z = 6? Then Y must be odd. */
otherwise if @149.z then if y//2 then iterate /*Z=1,4,9? Y must be even*/
end /*select*/
s= n+r; parse var s -2 y +1 z /*obtain the last two digits of the sum*/
if @ps.z then iterate /*Not 0, 2, 5, 8, or 9? Not perfect sq.*/
select
when z==0 then if y\==0 then iterate /*Does Z = 0? Then Y must be zero. */
when z==5 then if y\==2 then iterate /*Does Z = 5? Then Y must be two. */
when z==6 then if y//2==0 then iterate /*Does Z = 6? Then Y must be odd. */
otherwise if @149.z then if y//2 then iterate /*Z=1,4,9? Y must be even*/
end /*select*/
$= a + b /*a head start on figuring digital root*/
do k=3 for length(n) - 2 /*now, process the rest of the digits. */
$= $ + substr(n, k, 1) /*add the remainder of the digits in N.*/
end /*k*/
/*This REXX pgm uses 20 decimal digits.*/
do while $>9 /* [◄] Algorithm is good for 111 digs.*/
if $>9 then $= left($,1) + substr($,2,1)+ substr($,3,1,0) /*>9? Reduce to a dig*/
end /*while*/
if \@dr.$ then iterate /*Doesn't have good digital root? Skip*/
if iSqrt(s)**2 \== s then iterate /*Not a perfect square? Then skip it. */
if iSqrt(d)**2 \== d then iterate /* " " " " " " " */
#= # + 1 /*bump the counter of rare numbers. */
say right( th(#), length(#) + 9) ' rare number is: ' right( commas(n), w)
if #>=many then leave /* [↑] W: the width of # with commas.*/
end /*n*/
exit /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
commas: parse arg _; do jc=length(_)-3 to 1 by -3; _=insert(',', _, jc); end; return _
th: parse arg th;return th||word('th st nd rd',1+(th//10)*(th//100%10\==1)*(th//10<4))
/*──────────────────────────────────────────────────────────────────────────────────────*/
iSqrt: parse arg x; $= 0; q= 1; do while q<=x; q= q*4
end /*while q<=x*/
do while q>1; q= q % 4; _= x-$-q; $= $ % 2
if _>=0 then do; x= _; $= $ + q
end
end /*while q>1*/; return $</lang>
Still pretty sluggish, like molasses in March.
The above REXX program was modified to generate a group of numbers which were AB (two digit) numbers
concatenated with PQ (two digit) numbers to yield a list of four digit numbers.
AB are the 1st two digits of a rare number, and PQ are the last two digits.
This list was sorted and the duplicates removed, and it formed a list of (left 2 digits abutted with the right 2 digits)
numbers that every rare number must have (except for the first rare number (65), which is found the hard
(slow) way.