;# using TCLLIB math prime subroutines
             ;# COUNTING  SETS OF  10 WITH  PERCENTAGE  OF  PRIMES RELATIVE  TO  BASE_10
             puts "%| set_number:|percent of prime numbers in set | |%"
             foreach value {10 20 30 40 50 60 70 80 90 100 110 120} {
             puts "&| set $value :|[ expr {  [llength [ primesLowerThan [- $value 1 ] ]]*100./  [- $value 1 ]  }] |[ expr { ([llength [ primesLowerThan [- 10 1 ] ]]*100./  [- 10 1 ] )/( [llength [ primesLowerThan [- $value 1 ] ]]*100./  [- $value 1 ])  }]  |  |&" }

            puts "%|table 6 |printed in| tcl wiki format|% "
            puts "&| quantity| 60 / value| 1/ value|comment, if any|& "
            puts "&| entry |percent of prime numbers in set | |&"
            foreach value {1.81 2.08 2.33 2.09 } {
            puts "&| $value | [/ 60. $value ] |[/ 1. $value ]  |  |&" }
            foreach value {7 11 13 14 } {
            puts "&| $value | [/ 60. $value ] | [/ 1. $value ]  |  |&" } 

;# Babylonian algorithm of summing base_60 reciprocals
;# in using a false share
 set unit_price_1 [ expr { 60. / $market_rate_1 } ]
 set unit_price_2 [ expr { 60. / $market_rate_2 } ]
 set unit_price_3 [ expr { 60. / $market_rate_3 } ]
 set unit_price_4 [ expr { 60. / $market_rate_4 } ]
proc false_share2 {   available_silver  $unit_price_1    $unit_price_2    $unit_price_3   $unit_price_4  } {  return [ expr { $available_silver /  ($unit_price_1  +  $unit_price_2 +  $unit_price_3 + $unit_price_4 ) ] }

                c**2= a**2 + b**2
                c**2= a**2 + 1        normalized Babylonian triplet
                a**2  = c**2 -1       normalized Babylonian triplet
                some b and c are both primes.
                some b and c are not both primes.
                The reciprocal pair relationship.
                (X+1/X)**2 - (X+1/X)**2 = 4  ,    divide equation by 4 
                normalized triplet, < (1/4)* ((X+1/X)**2 ) , 1 ,(1/4)* ((X+1/X)**2) >
                0 < X-1/X < 2
                1< X-1/X < 1+sqrt(2)     ~~ approximate limit of 2.4 

       Triplet a,b,c & angle in degrees
       from tablet Plimpton 232
       prob. from Senkereh
       ref.E.C.  Zeeman
       119,120,169, 44.8
       3367,3456,4825,44.3
       4601,4800,6649,43.8
    ***12709,13500,18548,43.3***
       65,72,97,42.1
       319,360,481,41.5
       2291,2700,3541,40.3
       799,960,1249,39.8
       481,600,769,38.7
       4961,6480,8161,37.4
       3,4,5,37.4
       1679,2400,2929,35.0
       161,240,289,33.9
       1771,2700,3229,33.3
       28,45,53,31.9
       175,288,337,31.3 

        # adapted from tcl-wiki Stats 2011-05-22, arithmetic mean  [RLE]
        #
        # ::math::quantity_into_ratios --
        #
        # Return the division of quantity by  two or more given ratios
        #
        # Arguments:
        #    val  first value is quantity
        #    args other values are two or more given ratios
        #
        # Results: parts of quantity divided  by ratios
        #
        proc ::math::quantity_into_ratios {quantity args} {
            set sum 0.
            set N [ expr { [ llength $args ] + 1 } ]
            if { $N == 1  } { return 0 }
            foreach val $args {
                set sum [ expr { $sum + $val } ]
            }
            foreach val $args {
                lappend answer [ expr { $quantity * ($val/$sum) } ]
            }
            set answer
        }
        #puts "  ::math::quantity_into_ratios ( 84 2 2 2  )  answer 12.0 24.0 48.0    "
        #puts " [ ::math::quantity_into_ratios 84 1 2 4  ]  "
        #puts " [ ::math::quantity_into_ratios 84 ] for (::math::quantity_into_ratios 84) returns zero "

        # pretty print from autoindent and ased editor
        # easy eye calculator, large black type on green
        # used as testbed for one liners programs
        # written on Windows 10 on TCL
        # working under TCL version 8.6
        # gold on TCL WIKI , 10Sep2020
        package require Tk
        package require math::numtheory
        package require math::constants
        package require math::trig
        package require math
        namespace path {::tcl::mathop ::tcl::mathfunc math::numtheory math::trig math::constants }
        # adapted from tcl-wiki Stats 2011-05-22, arithmetic mean  [RLE]
        #
        #  sqrt_sum_of_squares --
        #
        # Return the sqrt_sum_of_squares by  one or more
        #
        # Arguments:
        #
        #    args other values are one or more
        #
        # Results: sqrt_sum_of_squares
        #
        proc math::sqrt_sum_of_squares { args} {
            set sum 0.
            set N [ expr { [ llength $args ] + 1 } ]
            if { $N == 1  } { return 0 }
            foreach val $args {
                set sum [ expr { $sum + $val*$val } ]
            }
            set answer [ expr { sqrt($sum) } ]
        }
        #puts "  ::math::sqrt_sum_of_squares (  2 2 2  )  answer [sqrt 12 ]
        #  3.4641016151377544
        #puts " [ ::math::sqrt_sum_of_squares   1 2 4  ]  "
        #puts " [ ::math::sqrt_sum_of_squares 2 ] for ( ) returns zero "
        console show
        console eval {.console config -bg palegreen}
        console eval {.console config -font {fixed 20 bold}}
        console eval {wm geometry . 40x20}
        console eval {wm title . "  Console wrapper for sqrt_sum_of_squares "}
        console eval {. configure -background orange -highlightcolor brown -relief raised -border 30}
        puts "  sqrt_sum_of_squares  2 2 2 equals [ math::sqrt_sum_of_squares  2 2 2 ]"
        puts "  sqrt_sum_of_squares  1 1 1 equals [ math::sqrt_sum_of_squares  1 1 1 ]"
        puts "  sqrt_sum_of_squares  1 1  equals [ math::sqrt_sum_of_squares  1 1  ]"
        puts "  sqrt_sum_of_squares  1  equals [ math::sqrt_sum_of_squares  1  ]"
        puts "  sqrt_sum_of_squares  0  equals [ math::sqrt_sum_of_squares  0  ]"

  sqrt_sum_of_squares  2 2 2 equals 3.4641016151377544
  sqrt_sum_of_squares  1 1 1 equals 1.7320508075688772
  sqrt_sum_of_squares  1 1  equals 1.4142135623730951
  sqrt_sum_of_squares  1  equals 1.0
  sqrt_sum_of_squares  0  equals 0.0

        # pretty print from autoindent and ased editor
        # quantity_into_ratios for math::stats? package
        # working under TCL version 8.6
        # console program written on Windows 10 on TCL
        # gold on TCL WIKI, 2Mar2017
        package require Tk
        console show
        package require math::numtheory
        namespace path {::tcl::mathop ::tcl::mathfunc math::numtheory }
        set tcl_precision 17
        # adapted from tcl-wiki Stats 2011-05-22, arithmetic mean  [RLE]
        #
        # ::math::quantity_into_ratios --
        #
        # Return the division of quantity by  two or more given ratios
        #
        # Arguments:
        #    val  first value is quantity
        #    args other values are two or more given ratios
        #
        # Results: parts of quantity divided  by ratios
        #    
        proc ::math::quantity_into_ratios {quantity args} {
            set sum 0.
            set N [ expr { [ llength $args ] + 1 } ]
            if { $N == 1  } { return 0 }
            foreach val $args {
                set sum [ expr { $sum + $val } ]
            }
            foreach val $args {
                lappend answer [ expr { $quantity * ($val/$sum) } ]
            }
            set answer
        }
        # end of working deck
        # add cosmetics below to bottom of file 
        console show
        console eval {.console config -bg palegreen}
        console eval {.console config -font {fixed 20 bold}}
        console eval {wm geometry . 40x20}
        console eval {wm title . "  Console wrapper for quantity_into_ratios  proc"}
        console eval {. configure -background orange -highlightcolor brown -relief raised -border 30}
        puts "  Console wrapper for quantity_into_ratios proc"
        puts " Return the quantities from   list of ratios"
        puts "  [ ::math::quantity_into_ratios  84 2 2 2 ] "  
        puts " for  ::math::quantity_into_ratios 84 2 2 2  returns 28.0 28.0 28.0 "
        puts " for  ::math::quantity_into_ratios 84 1 2 4 returns [ ::math::quantity_into_ratios 84 1 2 4  ] "
        puts "  for  ::math::quantity_into_ratios 84 1 2 4   returns 12.0 24.0 48.0 "    
        puts " [ ::math::quantity_into_ratios 84 ] <<  for  ::math::quantity_into_ratios 84  returns zero "      
        #end of deck

Console wrapper for quantity_into_ratios proc
 Return the quantities from   list of ratios
  28.0 28.0 28.0 
 for  ::math::quantity_into_ratios 84 2 2 2  returns 28.0 28.0 28.0 
 for  ::math::quantity_into_ratios 84 1 2 4 returns 12.0 24.0 48.0 
  for  ::math::quantity_into_ratios 84 1 2 4   returns 12.0 24.0 48.0 
 0 <<  for  ::math::quantity_into_ratios 84  returns zero 

     #pseudocode can be developed from rules of thumb.
     #pseudocode: some problems can be solved by proportions (rule of three), to some order of magnitude
     #pseudocode: enter quantity1,  quantity2, quantity3 and expected output (quantity4) for testcases.
     #pseudocode: enter time in years, number of remaining items
     #pseudocode: output fraction of (remaining items) over (items at time zero)
     #pseudocode: ouput remaining items as fraction or percent
     #pseudocode: output fraction of (quantity4 ) over ( quantity1 at time zero)
     #pseudocode: output fraction of (quantity2) * (quantity3 ) over (quantity1 at time zero)
     #pseudocode: outputs should be in compatible units.
     #pseudocode: rules of thumb can be 3 to 15 percent off, partly since g..in g..out.
     #pseudocode: need test cases > small,medium, giant
     #pseudocode: need testcases within range of expected operation.
     #pseudocode: are there any cases too small or large to be solved?
     

        # pretty print from autoindent and ased editor
        # geometric_mean for math::stats? package
        # working under TCL version 8.5.6 and eTCL 1.0.1
        # console program written on Windows XP on eTCL
        # gold on TCL WIKI, 2Mar2017
        package require Tk
        console show
        package require math::numtheory
        namespace path {::tcl::mathop ::tcl::mathfunc math::numtheory }
        set tcl_precision 17
        # adapted from tcl-wiki Stats 2011-05-22, arithmetic mean  [RLE]
        console show
        # ::math::geometric_mean --
        #
        # Return the geometric_mean of two or more values
        #
        # Arguments:
        #    val  first value
        #    args other values
        #
        # Results:
        #    geometric_mean  arithmetic geometric_mean value
        proc ::math::geometric_mean { args } {
            set product 1.
            set N [ expr { [ llength $args ] } ]
            foreach val $args {
                set product [ expr { $product*$val } ]
            }
            set exponent [ expr { 1./$N } ]
            set geometric_mean [ expr { $product**$exponent } ]
            #puts " $N   $exponent $product  $args "
            set geometric_mean
        }
        ;# end of working deck
        ;# add cosmetics below to bottom of file 
        console show
        console eval {.console config -bg palegreen}
        console eval {.console config -font {fixed 20 bold}}
        console eval {wm geometry . 40x20}
        console eval {wm title . "  Console wrapper for ::math::geometric_mean  proc"}
        console eval {. configure -background orange -highlightcolor brown -relief raised -border 30}
        puts "  Console wrapper for ::math::geometric_mean proc"
        puts " Return the quantities from   list of ratios"
        puts "  ::math::geometric_mean ( 1 2 3 4  )  answer  2.2133638  "
        puts " [ ::math::geometric_mean 1 2 3 4  ]   answer  2.2133638  "
        

                # pseudocode on Babylonian triplets
                2m, m**2 - 1, m**2 + 1  Babylonian triplet
                twin prime number.
                twin prime numbers separated by 2,4,6 ...?
                c**2= a**2 + b**2
                c**2= a**2 + 1        normalized Babylonian triplet
                a**2  = c**2 -1       normalized Babylonian triplet
                some b and c are both primes.
                some b and c are not both primes.
                The reciprocal pair relationship.
                (X+1/X)**2 - (X+1/X)**2 = 4  ,    divide equation by 4 
                normalized triplet, < (1/4)* ((X+1/X)**2 ) , 1 ,(1/4)* ((X+1/X)**2) >
                0 < X-1/X < 2
                1< X-1/X < 1+sqrt(2)     ~~ approximate limit of 2.4 
                The Babylonian regular numbers between 0 and 60 have a limited set of triplets that
                satisfy 1< X-1/X < 1+sqrt(2) which appears to be the role of Plimpton 322.
                The limit 1< X-1/X < 1+sqrt(2) may explain the 1<X<2 limit on the Late  Babylonian
                many place reciprocal tablets.
                d=6 even number, l = divisor of d**2 with integer result, Babylonian triplet method
                a=l+b
                b= d+ (d**2)//(2*l)
                c= d+l+(d**2)/(2*l)
                # supposed to work for all even d

                # Arguments:
                #     limit1            The limit for the largest prime to be included in the estimate
                #
                if { $limit1 <= 1 } {
                        return -code error "The limit must be larger than 1"
                }
                if { $limit2 <= 1 } {
                        return -code error "The upper limit must be larger than 1"
                }
                if { $n == 0 } {return 0}
                if { $n < 1  } {return 0}
                if { $n == 1 } {return 1}
                if {$n<2}      then {return -code return 2}
                if {$n<4}      then {return -code return 4}
                if {$n<9}      then {return -code return 9}
                #    forgivable error but no entry forwarded for calculation
                if { [ isprime $limit3 ] < 1 } {
                        break
                }
                # Add the additional procedures to TCLLIB::math local copy
                #
                proc ::math::numtheory::call {args} {uplevel catch [list $args]}
                proc ::math::numtheory::mathx { args } { set tcl_precision 17; puts [ expr [ expr { $args } ] ] }
                ;# translating Fortran and Qbasic control flow statements
                proc ::math::numtheory::stop {} { return }
                proc ::math::numtheory::end {}  { exit }
                proc ::math::numtheory::pause {} { wait .00010}
                ;# set comment lines from Fortran
                proc ::math::numtheory::C {args} { set C # }
                proc ::math::numtheory::REM {args}  { set REM # }     ;# attack of bots from Qbasic
                proc ::math::numtheory::write {args}  { puts $args }
                proc ::math::numtheory::print {args}  { puts $args }  ;# attack of bots from Quickbasic
                proc ::math::numtheory::SUBROUTINE {args}  { proc $args }
                proc ::math::numtheory::SUB  {args}  { proc $args }   ;# attack of bots from Quickbasic
                ;# namespace eval ::math::constants {
                variable constants {
                        pi        3.14159265358979323846   "ratio of circle circumference and diameter"
                        e         2.71828182845904523536   "base for natural logarithm"
                        ln10      2.30258509299404568402   "natural logarithm of 10"
                        phi       1.61803398874989484820   "golden ratio"
                        phi_2    0.61803398874989484820   "golden ratio conjugate"
                tcl_precision_1    17    "tcl_precision number"

                        )   set result {}
                        ;# set list for calculation results
                        lappend result $number
                }
                return $result

;# References 
;# current TCLLIB routines in Math Module
math::numtheory::numberPrimesGauss N
math::numtheory::numberPrimesLegendre N
math::numtheory::numberPrimesLegendreModified N
math::numtheory::differenceNumberPrimesLegendreModified lower upper
math::numtheory::listPrimePairs lower upper step
math::numtheory::listPrimeProgressions lower upper step

        # pretty print from autoindent and ased editor
        # easy eye format, large black type on green
        # used as testbed for one liners programs
        # written on Windows 10 on TCL
        # working under TCL version 8.6
        # gold on TCL WIKI , 5/15/2021
        package require Tk
        package require math::numtheory
        package require math::constants
        package require math::trig
        package require math
        namespace path {::tcl::mathop ::tcl::mathfunc math::numtheory math::trig math::constants }
        # adapted from tcl-wiki Stats 2011-05-22, arithmetic mean  [RLE]
        #
        #  prime_formula --
        #
        # Return the possible primes in list by  one or more
        #
        # Arguments:
        #
        #    args other values are one or more
        #
        # Results: prime_formula
        #
        proc prime_formula { args} {
            set list_new [list  ]
            set N [ expr { [ llength $args ] + 1 } ]
            if { $N == 1  } { return 0 }
            foreach val $args {
                lappend list_new [ expr {  16*$val + 15. } ]
            }
            return $list_new
        }
        # end of initial deck
        # add cosmetics below to bottom of file 
        console show
        console eval {.console config -bg palegreen}
        console eval {.console config -font {fixed 20 bold}}
        console eval {wm geometry . 40x20}
        console eval {wm title . "  Console wrapper for prime_formula proc"}
        console eval {. configure -background orange -highlightcolor brown -relief raised -border 30}
        puts "  Console wrapper for prime_formula proc"
        puts " Return the possible primes in list "
        puts " formula < 7 11 13 14 >   equals possible primes? [  prime_formula 7 11 13 14 ]"
                             

        ;# pretty print from autoindent and ased editor
        ;# easy eye console, large black type on green
        ;# used as testbed for one liners programs
        ;# written on Windows 10 on TCL
        ;# working under TCL version 8.6
        ;# gold on TCL WIKI , 10Sep2020
        ;# Babylonian Combined Market Rate V6
        package require Tk
        package require math::numtheory
        package require math::constants
        package require math::trig
        package require math
        namespace path {::tcl::mathop ::tcl::mathfunc math::numtheory math::trig math::constants }
        ;# adapted from tcl Stats 2011-05-22, arithmetic mean  <RLE>
        ;# ::math::combined_market_rate --
        ;#
        ;# Return the combined_market_rate by  one,two, or more given rates
        ;# market rate defined as quantity per price
        ;# or 1 over (price per quantity)
        ;#
        ;# Arguments:
        ;#
        ;#    args  values are one, two, or more given rates
        ;#
        ;# Results: combined_market_rate
        ;#  works for positive numbers, negative numbers,
        ;#  and mixed positive & negative numbers.
        ;#  arg of zero returns zero
        ;#  arg of null returns zero
        ;#  filter foreach drops irregular zero elements from argument
        proc ::math::combined_market_rate { args} {
            set sum 0.
            set N [ expr { [ llength $args ] } ]
            if { $N == 0 } { return 0 }
            if { $N == 1 || [ lindex $args 0 ] == 0 } { return 0 }
            set res {};set counter2 0;
            ;# filter foreach drops irregular zero elements
            foreach item $args {if {$item != 0 } {incr counter2 1; lappend res $item } }
            set counter 0
            foreach val $res {
                set sum [ expr { $sum + 1./$val } ]
                incr counter 1
            }
            set combined_market_rate1 [ expr { 1./(($sum*1.)/$counter2) } ]
            return $combined_market_rate1
        }
        ;# various testcases on combined_market_rate
        ;# puts  [::math::combined_market_rate  0.5524 0.4807 0.42918 0.47846 ]
        ;#  answer 0.48131    "
        ;# puts   [ ::math::combined_market_rate .1 .2 .3 .4 ]
        ;#  answer 0.192
        ;# ::math::combined_market_rate -.1 -.2 -.3 -.4
        ;# answer -0.192, correct
        ;# operator math formula follows
        ;# check [/ 1. [/ [+ [/ 1. -.1] [/ 1. -.2] [/ 1. -0.3 ] [/ 1. -0.4] ] 4. ] ]
        ;# returns  -0.192, correct
        ;# puts " [ ::math::combined_market_rate .1 ] "
        ;# :math::combined_market_rate -.1 -.2  .3 .4
        ;# answer -0.4363636363636364
        ;# operator math formula follows
        ;# set check [/ 1. [/ [+ [/ 1. -.1] [/ 1. -.2] [/ 1.  0.3 ] [/ 1. 0.4] ] 4. ] ]
        ;# check equals    -0.4363636363636364
        ;# puts " for (::math::combined_market_rates .1)
        ;# returns .1 "
        ;# ::math::combined_market_rate  {}
        ;# null returns zero, correct.
        ;# ::math::combined_market_rate 0
        ;# arg 0 returns zero, correct.
        ;# addition dated 24sep2018
        ;# added filter foreach to remove zero's
        ;# irregular zeros,
        ;# test on zero's 0.1 0.0 0.0 0.2 0.3 0.4
        ;# returns 0.192 , correct
        ;# test on zero's   0.1 0.0 0.0 0.2 0.3 0.4
        ;# returns 0.192 , correct      
        # end of initial deck
        # add cosmetics below to bottom of file 
        console show
        console eval {.console config -bg palegreen}
        console eval {.console config -font {fixed 20 bold}}
        console eval {wm geometry . 40x20}
        console eval {wm title . "  Console wrapper for combined market rate proc"}
        console eval {. configure -background orange -highlightcolor brown -relief raised -border 30}
        puts "  Console wrapper for combined market rate proc"
        puts " Return the combined market rate from  list "
        puts " formula < 0.5524 0.4807 0.42918 0.47846  >   equals market rate? [ ::math::combined_market_rate  0.5524 0.4807 0.42918 0.47846 ]"
        puts " formula <  7. 11. 13. 14.  >   equals market rate? [ ::math::combined_market_rate 7. 11. 13. 14.  ]"