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

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, 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, 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

Offset

1,1

Comments

See A204892 for a discussion and guide to related sequences.

Links

Antti Karttunen, Table of n, a(n) for n = 1..6556

Formula

a(n) = max(1, A007814(n)) + A007733(n). - Andrew Howroyd, Aug 08 2018

Example

1 divides 2^2 - 2^1, so a(1)=2;
2 divides 2^2 - 2^1, so a(2)=2;
3 divides 2^3 - 2^1, so a(3)=3;
4 divides 2^3 - 2^2, so a(4)=3;
5 divides 2^5 - 2^1, so a(5)=5.

Mathematica

s[n_] := s[n] = 2^n; z1 = 1000; z2 = 50;
Table[s[n], {n, 1, 30}]     (* A000079 *)
u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]
Table[u[m], {m, 1, z1}]     (* A204985 *)
v[n_, h_] := v[n, h] = If[IntegerQ[u[h]/n], h, 0]
w[n_] := w[n] = Table[v[n, h], {h, 1, z1}]
d[n_] := d[n] = First[Delete[w[n], Position[w[n], 0]]]
Table[d[n], {n, 1, z2}]     (* A204986 *)
k[n_] := k[n] = Floor[(3 + Sqrt[8 d[n] - 1])/2]
m[n_] := m[n] = Floor[(-1 + Sqrt[8 n - 7])/2]
j[n_] := j[n] = d[n] - m[d[n]] (m[d[n]] + 1)/2
Table[k[n], {n, 1, z2}]     (* A204987 *)
Table[j[n], {n, 1, z2}]     (* A204988 *)
Table[s[k[n]], {n, 1, z2}]  (* A204989 *)
Table[s[j[n]], {n, 1, z2}]  (* A140670 ? *)
Table[s[k[n]] - s[j[n]], {n, 1, z2}]     (* A204991 *)
Table[(s[k[n]] - s[j[n]])/n, {n, 1, z2}] (* A204992 *)
%%/2  (* A204990=(1/2)*A204991 *)

Prog

(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
(PARI) a(n)={my(k=valuation(n, 2)); max(k, 1) + znorder(Mod(2, n>>k))} \\ Andrew Howroyd, Aug 08 2018

Crossrefs

Cf. A000079, A007733, A007814, A054703, A204892, A204988.

Sequence in context: A254309 A079375 A069933 * A102347 A230846 A204908

Adjacent sequences: A204984 A204985 A204986 * A204988 A204989 A204990

Keyword

nonn

Author

Clark Kimberling, Jan 21 2012

Extensions

More terms from Antti Karttunen, Nov 19 2017

Status

approved