A333600 a(n) is the greatest possible length of a list of pairwise coprime distinct positive integers in arithmetic progression with greatest element n.
1, 2, 3, 2, 3, 2, 4, 2, 3, 2, 4, 2, 5, 2, 3, 2, 5, 2, 5, 2, 3, 2, 5, 2, 5, 2, 3, 2, 5, 2, 6, 2, 3, 2, 5, 2, 7, 2, 3, 2, 6, 2, 8, 2, 3, 2, 7, 2, 7, 2, 3, 2, 8, 2, 5, 2, 3, 2, 9, 2, 6, 2, 3, 2, 5, 2, 7, 2, 3, 2, 6, 2, 8, 2, 3, 2, 7, 2, 9, 2, 3, 2, 8, 2, 5, 2, 3
Offset: 1
Examples
The first terms, alongside a corresponding list, are: n a(n) List -- ---- ---- 1 1 (1) 2 2 (1, 2) 3 3 (1, 2, 3) 4 2 (3, 4) 5 3 (1, 3, 5) 6 2 (5, 6) 7 4 (1, 3, 5, 7) 8 2 (7, 8) 9 3 (7, 8, 9) 10 2 (9, 10) 11 4 (5, 7, 9, 11) 12 2 (11, 12) 13 5 (5, 7, 9, 11, 13)
Links
- Robert Israel, Table of n, a(n) for n = 1..10000
- Wikipedia, Green-Tao theorem
Programs
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Maple
f:= proc(n) local d, m, p, x, mmax; if n::even then return 2 fi; if n mod 3 = 0 then return 3 fi; mmax:= 1; for d from 1 to n-1 do if n <= mmax*d then return mmax fi; p:= n; for m from 1 to n/d do x:= n - d*m; if igcd(x,p) > 1 then break fi; p:= p*x; od; mmax:= max(mmax, m) od; end proc: f(1):= 1: map(f, [$1..100]); # Robert Israel, Apr 03 2020 -
Mathematica
a[n_] := Module[{d, m, p, x, mmax}, If[EvenQ[n], Return[2]]; If[Mod[n, 3] == 0, Return[3]]; mmax = 1; For[d = 1, d <= n-1, d++, If[n <= mmax d, Return[mmax]]; p = n; For[m = 1, m <= n/d, m++, x = n - d m; If[GCD[x, p] > 1, Break[]]; p = p x]; mmax = Max[mmax, m]]]; a[1] = 1; Array[a, 100] (* Jean-François Alcover, Oct 25 2020, after Robert Israel *) -
PARI
a(n) = { if (n%2==0, return (2), my (v=1); for (s=1, n-1, if (v>=ceil(n/s), break); my (p=1, w=0); forstep (k=n, 1, -s, if (gcd(p,k)==1, p*=k; w++, break)); v=max(v,w)); return (v)) }
Formula
a(2*n) = 2 for any n > 0.
a(prime(n)) > A109831(n) for any n > 1.
a(n) <= A020639(n) for n > 1. - Robert Israel, Apr 03 2020
Comments