%I A089587
%S A089587 0,1,2,3,2,5,4,3,7,8,3,5,9,4,8,11,2,13,11,3,16,5,6,5,21,9,19,18,16,8,16,
%T A089587 23,9,6,16,31,26,11,20,23,16,11,36,27,34,12,6,29,43,41,26,35,16,37,43,
%U A089587 25,49,49,3,53,47,16,32,57,38,4,37,39,12,16,57,61,37,26,61,49,67,47,55
%N A089587 a(n) is the smallest integer k, 0<k<n, that most often satisfies the 
               condition: k^m > k^(m+1) (modulo n) as m varies from 1 to n-1, for 
               n>2, with a(1)=0 and a(2)=1.
%C A089587 The values at a(2^n+1), n>=0, are given by A089588: {1,2,2,7,2,9,38,79,
               2,220,..} and it is conjectured that a(n)=2 only when n is a Fermat 
               number 2^(2^j)+1.
%F A089587 a(2^(2^n)+1)=2; a(2^n-1)=2^(n-1); a(4^n-2)=4^(n-1) (conjectures).
%e A089587 a(7)=4 since 4 is the smallest number between 1 and 6 that has the maximum 
               number of decreasing power residues mod 7:
%e A089587 n=7: k=1..6, m=1..7
%e A089587 k^m (mod 7).. #>'s
%e A089587 1=1=1=1=1=1=1:0
%e A089587 2<4>1<2<4>1<2:2
%e A089587 3>2<6>4<5>1<3:3
%e A089587 4>2>1<4>2>1<4:4 <- max of 4 >'s first occurs at k=4
%e A089587 5>4<6>2<3>1<5:3
%e A089587 6>1<6>1<6>1<6:3
%o A089587 (PARI) {a(n)=local(A); n>=0; M=0; A=1;
%o A089587 (PARI) a(n) = local(high, lowK, count); high = 0; for (k = 1, n - 1, 
               count = 0; for (m = 1, n - 1, if (k^m%n > k^(m+1)%n, count++)); if 
               (count > high, high = count; lowK = k)); lowK; (David Wasserman (wasserma(AT)spawar.navy.mil), 
               Sep 20 2005)
%Y A089587 Cf. A089588, A000215.
%Y A089587 Sequence in context: A132226 A075861 A141658 this_sequence A067316 A127433 
               A055573
%Y A089587 Adjacent sequences: A089584 A089585 A089586 this_sequence A089588 A089589 
               A089590
%K A089587 nonn
%O A089587 1,3
%A A089587 Paul D. Hanna (pauldhanna(AT)juno.com), Nov 09 2003