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A226182 a(n) is the smallest integer k >= 2 such that the number of divisors d>1 of n + k with k|n + d equals A225867(n).

%I A226182 #15 Jul 04 2013 15:01:55
%S A226182 2,2,2,3,2,2,4,2,2,3,2,3,2,2,4,3,2,2,3,5,2,3,2,3,2,2,4,3,2,5,4,3,2,3,
%T A226182 2,3,2,2,4,3,2,2,2,7,2,3,2,2,2,5,4,3,2,4,4,2,2,3,2,5,6,3,4,3,2,4,8,2,
%U A226182 2,5,4,7,2,2,4,3,2,2,4,5,2,3,2,2,6,7,4
%N A226182 a(n) is the smallest integer k >= 2 such that the number of divisors d>1 of n + k with k|n + d equals A225867(n).
%H A226182 Peter J. C. Moses, <a href="/A226182/b226182.txt">Table of n, a(n) for n = 6..3005</a>
%e A226182 Let n = 33. We begin with k = 2. Divisors>1 of 33 + 2 = 35 are d = 5,7,35. For all d, 33 + d is divisible by k = 2. But the number of such d is 3, while A225867(33)= 6. Therefore, a(33) > 2. Consider now k = 3. Divisors>1 of 33 + 3 = 36 are 2,3,4,6,9,12,18,36, but only for d = 3,6,9,12,18,36, 33 + d is divisible by k = 3. Since we have exactly A225867(33) = 6 such divisors, then k = 3 is required and a(33) = 3.
%p A226182 A226182 := proc(n)
%p A226182     local ak,k,nd,kpiv ;
%p A226182     ak := 0 ;
%p A226182     kpiv := 2 ;
%p A226182     for k from 2 to n/2-1 do
%p A226182         nd := 0 ;
%p A226182         for d in numtheory[divisors](n+k) minus {1} do
%p A226182             if modp(n+d,k) = 0 then
%p A226182                 nd := nd+1;
%p A226182             end if;
%p A226182         end do:
%p A226182         if nd > ak then
%p A226182             ak := max(ak,nd) ;
%p A226182             kpiv := k ;
%p A226182         end if;
%p A226182     end do:
%p A226182     kpiv ;
%p A226182 end proc: # _R. J. Mathar_, Jul 04 2013
%t A226182 Table[NestWhile[#+1&, 2, Max[Map[Count[(n+Rest[Divisors[n+#]])/#, _Integer]&, Range[2, Floor[(n-2)/2]]]]-Count[(n+Rest[Divisors[n+#]])/#, _Integer] =!= 0&], {n,6,55}] (* _Peter J. C. Moses_, Jun 03 2013 *)
%Y A226182 Cf. A225867, A225868, A188550, A188794.
%K A226182 nonn
%O A226182 6,1
%A A226182 _Vladimir Shevelev_, May 30 2013