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A270096 Smallest m such that 2^m == 2^n (mod n).

%I A270096 #53 Aug 03 2017 11:53:35
%S A270096 0,1,1,2,1,2,1,3,3,2,1,2,1,2,3,4,1,6,1,4,3,2,1,4,5,2,9,4,1,2,1,5,3,2,
%T A270096 11,6,1,2,3,4,1,6,1,4,9,2,1,4,7,10,3,4,1,18,15,5,3,2,1,4,1,2,3,6,5,6,
%U A270096 1,4,3,10,1,6,1,2,15,4,17,6,1,4
%N A270096 Smallest m such that 2^m == 2^n (mod n).
%C A270096 a(n) = 1 iff n is a prime or a pseudoprime (odd or even) to base 2.
%C A270096 We have a(n) <= n - phi(n) and a(n) <= phi(n), so a(n) <= n/2.
%C A270096 From _Robert Israel_, Mar 11 2016: (Start)
%C A270096 If n is in A167791, then a(n) = A068494(n).
%C A270096 If n is odd, a(n) = n mod A002326((n-1)/2).
%C A270096 a(n) >= A007814(n).
%C A270096 a(p^k) = p^(k-1) for all k >= 1 and all odd primes p not in A001220.
%C A270096 Conjecture: a(n) <= n/3 for all n > 8. (End)
%H A270096 Robert Israel, <a href="/A270096/b270096.txt">Table of n, a(n) for n = 1..10000</a>
%F A270096 a(n) < n/2 for n > 4.
%F A270096 a(2^k) = k for all k >= 0.
%F A270096 a(2*p) = 2 for all primes p.
%p A270096 f:= proc(n) local d,b,t, m,c;
%p A270096   d:= padic:-ordp(n,2);
%p A270096   b:= n/2^d;
%p A270096   t:= 2 &^ n mod n;
%p A270096   m:= numtheory:-mlog(t,2,b,c);
%p A270096   if m < d then m:= m + c*ceil((d-m)/c) fi;
%p A270096   m
%p A270096 end proc:
%p A270096 f(1):= 0:
%p A270096 map(f, [$1..1000]; # _Robert Israel_, Mar 11 2016
%t A270096 Table[k = 0; While[PowerMod[2, n, n] != PowerMod[2, k, n], k++]; k, {n, 120}] (* _Michael De Vlieger_, Mar 15 2016 *)
%o A270096 (PARI) a(n) = {my(m = 0); while (Mod(2, n)^m != 2^n, m++); m; } \\ _Altug Alkan_, Sep 23 2016
%Y A270096 Cf. A000010, A001220, A002326, A007814, A051953, A068494, A167791.
%Y A270096 Cf. A276976 (a generalization on all integer bases).
%K A270096 nonn
%O A270096 1,4
%A A270096 _Thomas Ordowski_, Mar 11 2016
%E A270096 More terms from _Michel Marcus_, Mar 11 2016