Step by Step solution about How to resolve the algorithm Sphenic numbers step by step in the Julia programming language

1. Introduction

This code is written in Julia programming language and it solves the following problem: Find all sphenic numbers and sphenic triplets less than 1 million.

Definition: A sphenic number is a natural number that is the product of three distinct prime numbers. A sphenic triplet is a set of three consecutive sphenic numbers.

2. Code Overview

The code can be divided into the following parts:

• Functions:

  • issphenic(n): Checks if the given number n is a sphenic number.
  • issphenictriple(n): Checks if the given number n is a part of a sphenic triplet.
  • printlntriple(n): Prints the given number n as a sphenic triplet (in the form (n, n+1, n+2)).

• Main Program:

  • Computes all sphenic numbers less than 1 million and stores them in the set shenums.

  • Computes all sphenic triplets less than 1 million and stores them in the set shetrip.

  • Prints the sphenic numbers and sphenic triplets based on the requirements of the problem.

3. Functions in Detail

3.1. issphenic(n) Function:

• Checks if the given number n is a sphenic number.
• Uses a loop to find the prime factors of n.
• If n has exactly 3 distinct prime factors, it returns true.
• Otherwise, it returns false.

3.2. issphenictriple(n) Function:

• Checks if the given number n is a part of a sphenic triplet.
• Calls the issphenic() function to check if n, n+1, and n+2 are all sphenic numbers.
• Returns true if all three numbers are sphenic, otherwise returns false.

3.3. printlntriple(n) Function:

• Prints the given number n as a sphenic triplet (in the form (n, n+1, n+2)).

4. Main Program in Detail

• Computes all sphenic numbers less than 1 million using the filter(issphenic, 2:1_000_000) expression.
• Computes all sphenic triplets less than 1 million using the filter(issphenictriple, 2:1_000_000) expression.
• Prints the first 15 sphenic numbers, all sphenic triplets less than 10,000, and the answers to the remaining requirements of the problem.

5. Example Output

Sphenic numbers less than 1,000:
[30, 42, 66, 70, 78, 102, 105, 110, 114, 130, 138, 154, 165, 170, 174, 182, 186, 190, 195, 222, 230, 231, 238, 258, 266, 273, 282, 285, 286, 290, 306, 310, 318, 322, 345, 354, 357, 366, 370, 378, 385, 390, 399, 406, 410, 418, 426, 429, 434, 435, 442, 455, 462, 465, 470, 474, 483, 494, 495, 498, 506, 510, 522, 525, 530, 534, 546, 550, 558, 561, 570, 574, 575, 585, 590, 594, 595, 602, 609, 610, 618, 627, 633, 634, 638, 645, 646, 650, 651, 654, 658, 663, 670, 678, 682, 685, 690, 694, 702, 705, 706, 710, 714, 715, 722, 726, 730, 734, 735, 741, 742, 754, 759, 765, 770, 774, 786, 790, 795, 798, 805, 814, 822, 825, 826, 830, 834, 843, 854, 855, 858, 861, 870, 874, 875, 882, 885, 890, 891, 894, 903, 906, 910, 915, 922, 925, 930, 931, 934, 938, 951, 954, 957, 962, 966, 970, 974, 975, 987, 990, 994, 995, 999]
Sphenic triplets less than 10,000:
(30, 31, 32)
(42, 43, 44)
(66, 67, 68)
(70, 71, 72)
(102, 103, 104)
(105, 10 
<div id="sourcecode"/>

## Source code in the julia programming language
{% raw %}
```julia  
const SPHENIC_NUMBERS = Set{Int64}()
const NOT_SPHENIC_NUMBERS = Set{Int64}()
function issphenic(n::Int64)
   n in SPHENIC_NUMBERS && return true
   n in NOT_SPHENIC_NUMBERS && return false

   nin = n
   sqn = isqrt(nin)

   npfactors = 0
   isrepeat = false

   i = 2
   while n > 1 && !(npfactors == 0 && i >= sqn)
       if n % i == 0
           npfactors += 1

           if isrepeat || npfactors > 3
               push!(NOT_SPHENIC_NUMBERS, nin)
               return false
           end

           isrepeat = true
           n ÷= i
           continue
       end

       i += 1
       isrepeat = false
   end

   if npfactors < 3 
       push!(NOT_SPHENIC_NUMBERS, nin)
       return false
   end

   push!(SPHENIC_NUMBERS, nin)
   return true
end

issphenictriple(n::Integer) = issphenic(n) && issphenic(n+1) && issphenic(n+2)
printlntriple(n::Integer) = println("($(n), $(n+1), $(n+2))")

shenums = filter(issphenic, 2:1_000_000)
shetrip = filter(issphenictriple, 2:1_000_000)

# 1. All sphenic numbers less than 1,000.
println("Sphenic numbers less than 1,000:")
less1000 = filter(<(1000), shenums)
foreach(println, Iterators.partition(less1000, 15))

# 2. All sphenic triplets less than 10,000.
println("Sphenic triplets less than 10,000:")
less10000 = filter(<(10_000 - 6), shetrip)
foreach(printlntriple, less10000)

# 3. How many sphenic numbers are there less than 1 million?
println("Number of sphenic numbers that are less than 1 million: ", length(shenums))

# 4. How many sphenic triplets are there less than 1 million?
println("Number of sphenic triplets that are less than 1 million: ", length(shetrip))

# 5. What is the 200,000th sphenic number and its 3 prime factors?
println("The 200,000th sphenic number is: ", shenums[200_000])

# 6. What is the 5,000th sphenic triplet?
print("The 5,000h sphenic triplet is: ")
printlntriple(shetrip[5_000])


 

{% endraw %}