Erdös-Selfridge categorization of primes

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