Rare numbers
Rare numbers are positive integers n where:
- Definitions and restrictions
- n is expressed in base ten
- n must be non-palindromic ( n ≠ r)
- r is the reverse of n (decimal digits)
- (n+r) is the sum
- (n-r) is the difference and must be positive
- the sum and the difference must be perfect squares
- Task
-
- find and show the first 5 rare numbers
- find and show the first 6 rare numbers (optional)
- find and show more rare numbers (stretch goal)
Show all output here, on this page.
- References
-
- an OEIS entry: A035519 rare numbers.
- an OEIS entry: A059755 odd rare numbers.
- planetmath entry: rare numbers. (some hints)
- author's website: rare numbers by Shyam Sunder Gupta. (lots of hints and some observations).
REXX
<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 must'nt 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 must'nt 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>
- output when using the input of: 8
1st rare number is: 65
2nd rare number is: 621,770
3rd rare number is: 281,089,082
4th rare number is: 2,022,652,202
5th rare number is: 2,042,832,002
6th rare number is: 868,591,084,757
7th rare number is: 872,546,974,178
8th rare number is: 872,568,754,178