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).
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, 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, 2, 5, 4, 7, 2, 2, 4, 3, 2, 2, 4, 5, 2, 3, 2, 2, 6, 7, 4
Offset: 6
Keywords
Examples
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.
Links
- Peter J. C. Moses, Table of n, a(n) for n = 6..3005
Programs
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Maple
A226182 := proc(n) local ak,k,nd,kpiv ; ak := 0 ; kpiv := 2 ; for k from 2 to n/2-1 do nd := 0 ; for d in numtheory[divisors](n+k) minus {1} do if modp(n+d,k) = 0 then nd := nd+1; end if; end do: if nd > ak then ak := max(ak,nd) ; kpiv := k ; end if; end do: kpiv ; end proc: # R. J. Mathar, Jul 04 2013
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Mathematica
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 *)
Comments