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A293273 a(n) is the smallest positive k <> n such that f(k) is divisible by f(n) where f = A005132, or 0 if no such k exists.

%I A293273 #36 Oct 20 2017 23:11:40
%S A293273 2,3,8,3,9,35,43,15,20,11,28,7,32,21,83,15,69,26,152,24,116,47,44,20,
%T A293273 48,18,43,59,30,63,20,104,41,71,39,75,72,35,35,36,33,79,92,83,96,87,
%U A293273 100,91,245,95,239,67,276,19,119,63,109,57,103,51,185,45,139,35,145,86,415,84,192,82,184,80,180,78,176
%N A293273 a(n) is the smallest positive k <> n such that f(k) is divisible by f(n) where f = A005132, or 0 if no such k exists.
%C A293273 Conjecture: a(n) > 0 for all n.
%H A293273 Robert Israel, <a href="/A293273/b293273.txt">Table of n, a(n) for n = 1..10000</a>
%H A293273 Rémy Sigrist, <a href="/A293273/a293273.png">Logarithmic scatterplot of the first 100000 terms</a>
%H A293273 Rémy Sigrist, <a href="/A293273/a293273_1.png">Scatterplot of the first 100000 terms of the first difference</a>
%e A293273 a(6) = 35 because A005132(35) = 78 is divisible by A005132(6) = 13 and 78 is the smallest positive number which is not equal to 6 with this property.
%p A293273 N:= 10^4: # to use A005132(n) for n = 1..N
%p A293273 S:= {0}:
%p A293273 A5132:= Array(0..N):
%p A293273 A5132[0]:= 0:
%p A293273 for n from 1 to N do
%p A293273   v:= A5132[n-1]-n;
%p A293273   if v < 0 or member(v,S) then v:= A5132[n-1]+n fi;
%p A293273   A5132[n]:= v;
%p A293273   S:= S union {v};
%p A293273 od:
%p A293273 f:= proc(n) local k;
%p A293273   for k from 1 to N do
%p A293273     if k <> n and A5132[k] mod A5132[n] = 0 then return k fi
%p A293273   od:
%p A293273 0
%p A293273 end proc:
%p A293273 Res:= NULL:
%p A293273 for n from 1 do
%p A293273   v:= f(n);
%p A293273   if v = 0 then break fi;
%p A293273   Res:= Res,v;
%p A293273 od:
%p A293273 Res; # _Robert Israel_, Oct 10 2017
%Y A293273 Cf. A005132, A057167.
%K A293273 nonn,easy
%O A293273 1,1
%A A293273 _Altug Alkan_, Oct 10 2017