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A204987 Least k such that n divides 2^k - 2^j for some j satisfying 1 <= j < k.

%I A204987 #20 Aug 09 2018 03:26:00
%S A204987 2,2,3,3,5,3,4,4,7,5,11,4,13,4,5,5,9,7,19,6,7,11,12,5,21,13,19,5,29,5,
%T A204987 6,6,11,9,13,8,37,19,13,7,21,7,15,12,13,12,24,6,22,21,9,14,53,19,21,6,
%U A204987 19,29,59,6,61,6,7,7,13,11,67,10,23,13,36,9,10,37,21,20,31,13,40,8,55,21,83,8,9,15,29,13
%N A204987 Least k such that n divides 2^k - 2^j for some j satisfying 1 <= j < k.
%C A204987 See A204892 for a discussion and guide to related sequences.
%H A204987 Antti Karttunen, <a href="/A204987/b204987.txt">Table of n, a(n) for n = 1..6556</a>
%F A204987 a(n) = max(1, A007814(n)) + A007733(n). - _Andrew Howroyd_, Aug 08 2018
%e A204987 1 divides 2^2 - 2^1, so a(1)=2;
%e A204987 2 divides 2^2 - 2^1, so a(2)=2;
%e A204987 3 divides 2^3 - 2^1, so a(3)=3;
%e A204987 4 divides 2^3 - 2^2, so a(4)=3;
%e A204987 5 divides 2^5 - 2^1, so a(5)=5.
%t A204987 s[n_] := s[n] = 2^n; z1 = 1000; z2 = 50;
%t A204987 Table[s[n], {n, 1, 30}]     (* A000079 *)
%t A204987 u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]
%t A204987 Table[u[m], {m, 1, z1}]     (* A204985 *)
%t A204987 v[n_, h_] := v[n, h] = If[IntegerQ[u[h]/n], h, 0]
%t A204987 w[n_] := w[n] = Table[v[n, h], {h, 1, z1}]
%t A204987 d[n_] := d[n] = First[Delete[w[n], Position[w[n], 0]]]
%t A204987 Table[d[n], {n, 1, z2}]     (* A204986 *)
%t A204987 k[n_] := k[n] = Floor[(3 + Sqrt[8 d[n] - 1])/2]
%t A204987 m[n_] := m[n] = Floor[(-1 + Sqrt[8 n - 7])/2]
%t A204987 j[n_] := j[n] = d[n] - m[d[n]] (m[d[n]] + 1)/2
%t A204987 Table[k[n], {n, 1, z2}]     (* A204987 *)
%t A204987 Table[j[n], {n, 1, z2}]     (* A204988 *)
%t A204987 Table[s[k[n]], {n, 1, z2}]  (* A204989 *)
%t A204987 Table[s[j[n]], {n, 1, z2}]  (* A140670 ? *)
%t A204987 Table[s[k[n]] - s[j[n]], {n, 1, z2}]     (* A204991 *)
%t A204987 Table[(s[k[n]] - s[j[n]])/n, {n, 1, z2}] (* A204992 *)
%t A204987 %%/2  (* A204990=(1/2)*A204991 *)
%o A204987 (PARI) A204987etA204988(n) = { my(k=2); while(1,for(j=1,k-1,if(!(((2^k)-(2^j))%n),return([k,j]))); k++); }; \\ (Computes also A204988 at the same time) - _Antti Karttunen_, Nov 19 2017
%o A204987 (PARI) a(n)={my(k=valuation(n,2)); max(k, 1) + znorder(Mod(2, n>>k))} \\ _Andrew Howroyd_, Aug 08 2018
%Y A204987 Cf. A000079, A007733, A007814, A054703, A204892, A204988.
%K A204987 nonn
%O A204987 1,1
%A A204987 _Clark Kimberling_, Jan 21 2012
%E A204987 More terms from _Antti Karttunen_, Nov 19 2017