How to resolve the algorithm EKG sequence convergence step by step in the Nim programming language

Published on 12 May 2024 09:40 PM
#Nim

How to resolve the algorithm EKG sequence convergence step by step in the Nim programming language

Table of Contents

Problem Statement

The sequence is from the natural numbers and is defined by: The sequence is called the EKG sequence (after its visual similarity to an electrocardiogram when graphed). Variants of the sequence can be generated starting 1, N where N is any natural number larger than one. For the purposes of this task let us call:

If an algorithm that keeps track of the minimum amount of numbers and their corresponding prime factors used to generate the next term is used, then this may be known as the generators essential state. Two EKG generators with differing starts can converge to produce the same sequence after initial differences. EKG(N1) and EKG(N2) are said to to have converged at and after generation a(c) if state_of(EKG(N1).a(c)) == state_of(EKG(N2).a(c)).

Let's start with the solution:

Step by Step solution about How to resolve the algorithm EKG sequence convergence step by step in the Nim programming language

Source code in the nim programming language

import algorithm, math, sets, strformat, strutils

#---------------------------------------------------------------------------------------------------

iterator ekg(n, limit: Positive): (int, int) =
  var values: HashSet[int]
  doAssert n >= 2
  yield (1, 1)
  yield (2, n)
  values.incl(n)
  var i = 3
  var prev = n
  while i <= limit:
    var val = 2
    while true:
      if val notin values and gcd(val, prev) != 1:
        values.incl(val)
        yield (i, val)
        prev = val
        break
      inc val
    inc i

#---------------------------------------------------------------------------------------------------

for n in [2, 5, 7, 9, 10]:
  var result: array[1..10, int]
  for i, val in ekg(n, 10): result[i] = val
  let title = fmt"EKG({n}):"
  echo fmt"{title:8} {result.join("", "")}"

var ekg5, ekg7: array[1..100, int]
for i, val in ekg(5, 100): ekg5[i] = val
for i, val in ekg(7, 100): ekg7[i] = val
var convIndex = 0
for i in 2..100:
  if ekg5[i] == ekg7[i] and sorted(ekg5[1..<i]) == sorted(ekg7[1..<i]):
    convIndex = i
    break
if convIndex > 0:
  echo fmt"EKG(5) and EKG(7) converge at index {convIndex}."
else:
  echo "No convergence found in the first {convIndex} terms."

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