Erdös-Selfridge categorization of primes: Difference between revisions

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say "First 200 primes; Erdös-Selfridge categorized:";
say "First 200 primes; Erdös-Selfridge categorized:";
my %category = $sieve.n-primes(200).categorize: { .&Erdös-Selfridge };
my %category = $sieve.n-primes(200).categorize: &Erdös-Selfridge;
.say for sort %category;
.say for sort %category;


say "\nFirst million primes; Erdös-Selfridge categorized:";
say "\nFirst million primes; Erdös-Selfridge categorized:";
%category = $sieve.n-primes(1_000_000).categorize: { .&Erdös-Selfridge };
%category = $sieve.n-primes(1_000_000).categorize: &Erdös-Selfridge;
printf "Category %2d: first: %7d, last: %8d, total: %d\n", ++$, .[0], .[*-1], .elems for %category.sort(+*.key)».value;</lang>
printf "Category %2d: first: %7d, last: %8d, total: %d\n", ++$, .[0], .[*-1], .elems for %category.sort(+*.key)».value;</lang>
{{out}}
{{out}}

Revision as of 02:37, 14 April 2022

Erdös-Selfridge categorization of primes is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

A prime p is in category 1 if the prime factors of p+1 are 2 and or 3. p is in category 2 if all the prime factors of p+1 are in category 1. p is in category g if all the prime factors of p+1 are in categories 1 to g-1.

The task is first to display the first 200 primes allocated to their category, then assign the first million primes to their category, displaying the smallest prime, the largest prime, and the count of primes allocated to each category.

F#

This task uses Extensible Prime Generator (F#) <lang fsharp> // Erdös-Selfridge categorization of primes. Nigel Galloway: April 12th., 2022 let rec fG n g=match n,g with ((_,1),_)|(_,[])->n |((_,p),h::_) when h>p->n |((p,q),h::_) when q%h=0->fG (p,q/h) g |(_,_::g)->fG n g let fN g=Seq.unfold(fun(n,g)->let n,g=n|>List.map(fun n->fG n g)|>List.partition(fun(_,n)->n<>1) in let g=g|>List.map fst in if g=[] then None else Some(g,(n,g)))(primes32()|>Seq.take g|>Seq.map(fun n->(n,n+1))|>List.ofSeq,[2;3]) fN 200|>Seq.iteri(fun n g->printfn "Category %d: %A" (n+1) g) fN 1000000|>Seq.iteri(fun n g->printfn "Category %d: first->%d last->%d count->%d" (n+1) (List.head g) (List.last g) ( </lang>

Output:
Category 1: [2; 3; 5; 7; 11; 17; 23; 31; 47; 53; 71; 107; 127; 191; 383; 431; 647; 863; 971;
 1151]
Category 2: [13; 19; 29; 41; 43; 59; 61; 67; 79; 83; 89; 97; 101; 109; 131; 137; 139; 149;
 167; 179; 197; 199; 211; 223; 229; 239; 241; 251; 263; 269; 271; 281; 283; 293;
 307; 317; 349; 359; 367; 373; 419; 433; 439; 449; 461; 479; 499; 503; 509; 557;
 563; 577; 587; 593; 599; 619; 641; 643; 659; 709; 719; 743; 751; 761; 769; 809;
 827; 839; 881; 919; 929; 953; 967; 991; 1019; 1033; 1049; 1069; 1087; 1103;
 1187; 1223]
Category 3: [37; 103; 113; 151; 157; 163; 173; 181; 193; 227; 233; 257; 277; 311; 331; 337;
 347; 353; 379; 389; 397; 401; 409; 421; 457; 463; 467; 487; 491; 521; 523; 541;
 547; 569; 571; 601; 607; 613; 631; 653; 683; 701; 727; 733; 773; 787; 797; 811;
 821; 829; 853; 857; 859; 877; 883; 911; 937; 947; 983; 997; 1009; 1013; 1031;
 1039; 1051; 1061; 1063; 1091; 1097; 1117; 1123; 1153; 1163; 1171; 1181; 1193;
 1217]
Category 4: [73; 313; 443; 617; 661; 673; 677; 691; 739; 757; 823; 887; 907; 941; 977; 1093;
 1109; 1129; 1201; 1213]
Category 5: [1021]

Category 1: first->2 last->10616831 count->46
Category 2: first->13 last->15482669 count->10497
Category 3: first->37 last->15485863 count->201987
Category 4: first->73 last->15485849 count->413891
Category 5: first->1021 last->15485837 count->263109
Category 6: first->2917 last->15485857 count->87560
Category 7: first->15013 last->15484631 count->19389
Category 8: first->49681 last->15485621 count->3129
Category 9: first->532801 last->15472811 count->363
Category 10: first->1065601 last->15472321 count->28
Category 11: first->8524807 last->8524807 count->1

Raku

<lang perl6>use Prime::Factor; use Math::Primesieve; my $sieve = Math::Primesieve.new;

sub Erdös-Selfridge ($n) { 

   my @factors = unique grep * > 3, prime-factors $n + 1;
   my $category = 1;
   while @factors.elems {
       @factors = unique grep * > 3, flat @factors.map: { prime-factors $_ + 1 };
       ++$category
   }
   $category

}

say "First 200 primes; Erdös-Selfridge categorized:"; my %category = $sieve.n-primes(200).categorize: &Erdös-Selfridge; .say for sort %category;

say "\nFirst million primes; Erdös-Selfridge categorized:"; %category = $sieve.n-primes(1_000_000).categorize: &Erdös-Selfridge; printf "Category %2d: first: %7d, last: %8d, total: %d\n", ++$, .[0], .[*-1], .elems for %category.sort(+*.key)».value;</lang>

Output:
First 200 primes; Erdös-Selfridge categorized:
1 => [2 3 5 7 11 17 23 31 47 53 71 107 127 191 383 431 647 863 971 1151]
2 => [13 19 29 41 43 59 61 67 79 83 89 97 101 109 131 137 139 149 167 179 197 199 211 223 229 239 241 251 263 269 271 281 283 293 307 317 349 359 367 373 419 433 439 449 461 479 499 503 509 557 563 577 587 593 599 619 641 643 659 709 719 743 751 761 769 809 827 839 881 919 929 953 967 991 1019 1033 1049 1069 1087 1103 1187 1223]
3 => [37 103 113 151 157 163 173 181 193 227 233 257 277 311 331 337 347 353 379 389 397 401 409 421 457 463 467 487 491 521 523 541 547 569 571 601 607 613 631 653 683 701 727 733 773 787 797 811 821 829 853 857 859 877 883 911 937 947 983 997 1009 1013 1031 1039 1051 1061 1063 1091 1097 1117 1123 1153 1163 1171 1181 1193 1217]
4 => [73 313 443 617 661 673 677 691 739 757 823 887 907 941 977 1093 1109 1129 1201 1213]
5 => [1021]

First million primes; Erdös-Selfridge categorized:
Category  1: first:       2, last: 10616831, total: 46
Category  2: first:      13, last: 15482669, total: 10497
Category  3: first:      37, last: 15485863, total: 201987
Category  4: first:      73, last: 15485849, total: 413891
Category  5: first:    1021, last: 15485837, total: 263109
Category  6: first:    2917, last: 15485857, total: 87560
Category  7: first:   15013, last: 15484631, total: 19389
Category  8: first:   49681, last: 15485621, total: 3129
Category  9: first:  532801, last: 15472811, total: 363
Category 10: first: 1065601, last: 15472321, total: 28
Category 11: first: 8524807, last:  8524807, total: 1

Wren

Library: Wren-math
Library: Wren-fmt

Second part takes a while. <lang ecmascript>import "./math" for Int import "./fmt" for Fmt

var limit = (1e6.log * 1e6 * 1.2).floor // should be more than enough var primes = Int.primeSieve(limit)

var cat // recursive cat = Fn.new { |p| 

   var pf = Int.primeFactors(p+1)
   if (pf.all { |f| f == 2 || f == 3 }) return 1
   for (c in 2..11) {
       if (pf.all { |f| cat.call(f) < c }) return c
   }
   return 12

}

var es = List.filled(12, null) for (i in 0..11) es[i] = []

System.print("First 200 primes:\n ") for (p in primes[0..199]) { 

   var c = cat.call(p)
   es[c-1].add(p)

} for (c in 1..6) { 

   if (es[c-1].count > 0) {
       System.print("Category %(c):")
       System.print(es[c-1])
       System.print()
   }

}

System.print("First million primes:\n") for (p in primes[200...1e6]) { 

   var c = cat.call(p)
   es[c-1].add(p)

} for (c in 1..12) { 

   var e = es[c-1]
   if (e.count > 0) {
       Fmt.print("Category $-2d: First = $7d  Last = $8d  Count = $6d", c, e[0], e[-1], e.count)
   }

}</lang>

Output:
First 200 primes:
 
Category 1:
[2, 3, 5, 7, 11, 17, 23, 31, 47, 53, 71, 107, 127, 191, 383, 431, 647, 863, 971, 1151]

Category 2:
[13, 19, 29, 41, 43, 59, 61, 67, 79, 83, 89, 97, 101, 109, 131, 137, 139, 149, 167, 179, 197, 199, 211, 223, 229, 239, 241, 251, 263, 269, 271, 281, 283, 293, 307, 317, 349, 359, 367, 373, 419, 433, 439, 449, 461, 479, 499, 503, 509, 557, 563, 577, 587, 593, 599, 619, 641, 643, 659, 709, 719, 743, 751, 761, 769, 809, 827, 839, 881, 919, 929, 953, 967, 991, 1019, 1033, 1049, 1069, 1087, 1103, 1187, 1223]

Category 3:
[37, 103, 113, 151, 157, 163, 173, 181, 193, 227, 233, 257, 277, 311, 331, 337, 347, 353, 379, 389, 397, 401, 409, 421, 457, 463, 467, 487, 491, 521, 523, 541, 547, 569, 571, 601, 607, 613, 631, 653, 683, 701, 727, 733, 773, 787, 797, 811, 821, 829, 853, 857, 859, 877, 883, 911, 937, 947, 983, 997, 1009, 1013, 1031, 1039, 1051, 1061, 1063, 1091, 1097, 1117, 1123, 1153, 1163, 1171, 1181, 1193, 1217]

Category 4:
[73, 313, 443, 617, 661, 673, 677, 691, 739, 757, 823, 887, 907, 941, 977, 1093, 1109, 1129, 1201, 1213]

Category 5:
[1021]

First million primes:

Category 1 : First =       2  Last = 10616831  Count =     46
Category 2 : First =      13  Last = 15482669  Count =  10497
Category 3 : First =      37  Last = 15485863  Count = 201987
Category 4 : First =      73  Last = 15485849  Count = 413891
Category 5 : First =    1021  Last = 15485837  Count = 263109
Category 6 : First =    2917  Last = 15485857  Count =  87560
Category 7 : First =   15013  Last = 15484631  Count =  19389
Category 8 : First =   49681  Last = 15485621  Count =   3129
Category 9 : First =  532801  Last = 15472811  Count =    363
Category 10: First = 1065601  Last = 15472321  Count =     28
Category 11: First = 8524807  Last =  8524807  Count =      1
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