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A138768 For a positive integer n, write the integers 1,2,...,n in the following order: first write 1 (round 0), then all primes less than or equal to n in increasing order (round 1), then 2p for all primes p with 2p<=n, also in increasing order (round 2), then 3p, then 4p and so on. Each number is written down only the first time it is encountered. Let a(n) denote the last number written down.

%I A138768 #12 Sep 05 2024 12:21:08
%S A138768 1,2,3,4,4,6,6,8,8,8,8,12,12,12,12,16,16,16,16,16,16,16,16,24,24,24,
%T A138768 27,27,27,27,27,32,32,32,32,32,32,32,32,32,32,32,32,32,32,32,32,48,48,
%U A138768 48,48,48,48,54,54,54,54,54,54,54,54,54,54,64,64,64,64,64,64,64,64,64,64
%N A138768 For a positive integer n, write the integers 1,2,...,n in the following order: first write 1 (round 0), then all primes less than or equal to n in increasing order (round 1), then 2p for all primes p with 2p<=n, also in increasing order (round 2), then 3p, then 4p and so on. Each number is written down only the first time it is encountered. Let a(n) denote the last number written down.
%C A138768 a(1)=1. For a given n>=2, let M be the largest of the numbers in the finite sequence [m/(largest prime dividing m), m=2,3,...,n]. a(n) is defined to be the largest m in (2,3,...,n) for which M is attained. Example: a(14)=12 because the values of m/(largest prime dividing m) for m = 2,3,...,14 are 1,1,2,1,2,1,4,3,2,1,4,1,2. The largest of these is 4 and it is attained for m=8 and m=12; the largest of these is 12.
%H A138768 Gary Gordon, <a href="https://www.jstor.org/stable/27642491">The Number between 1 and n That Is Least Prime: Problem 11218</a>, Amer. Math. Monthly, 115 (No. 4, 2008), pp. 367-368.
%e A138768 For n=10 we get the ordering 1/ 2, 3, 5, 7/ 4, 6, 10/ 9/ 8 (the rounds are separated by /); so a(10)=8.
%p A138768 with(numtheory): b:=proc(m) local u: if m=1 then 1 else u:=factorset(m): m/max(seq(u[j],j=1..nops(u))) end if end proc: a:=proc(n) local M,i,a: M:=max(seq(b(j),j=1..n)): for i to n do if b(i)=M then a[i]:=i else a[i]:=0 end if end do: max(seq(a[i],i=1..n)) end proc: seq(a(n),n=1..80);
%t A138768 b[m_] := If[m == 1, 1, m/Max[FactorInteger[m][[All, 1]]]];
%t A138768 a[n_] := Module[{M, i, a}, M = Max[Table[b[j], {j, 1, n}]]; For[i = 1, i <= n, i++, If[b[i] == M, a[i] = i, a[i] = 0]]; Max[Table[a[i], {i, 1, n}]]];
%t A138768 Table[a[n], {n, 1, 80}] (* _Jean-François Alcover_, Sep 05 2024, after Maple program *)
%Y A138768 Cf. A052126.
%K A138768 nonn
%O A138768 1,2
%A A138768 _Emeric Deutsch_ and _Gary Gordon_, Apr 01 2008