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A038026 Last position reached by winner of n-th Littlewood Frog Race.

%I A038026 #21 Jan 21 2022 23:28:47
%S A038026 2,3,7,5,19,7,29,17,19,19,43,13,103,29,31,41,103,19,191,41,67,43,137,
%T A038026 73,149,103,109,83,317,31,311,97,181,103,191,71,439,191,233,89,379,67,
%U A038026 463,113,181,137,967,97,613,149,197,181,607,109,331,233
%N A038026 Last position reached by winner of n-th Littlewood Frog Race.
%C A038026 Related to Linnik's theorem; main sequence is A085420. - _Charles R Greathouse IV_, Apr 16 2010
%C A038026 a(n) is the smallest prime such that some subset of primes <= a(n) is a reduced residue system modulo n. - _Vladimir Shevelev_, Feb 19 2013
%H A038026 Charles R Greathouse IV, <a href="/A038026/b038026.txt">Table of n, a(n) for n = 1..10000</a>
%F A038026 Let p(n,b) be the smallest prime in the arithmetic progression k*n+b, with k >= 0. Then a(n) = max(p(n,b)) with 0 < b < n and gcd(b,n) = 1. - _Charles R Greathouse IV_, Sep 08 2012
%e A038026 a(6) = 7 since the primes less than or equal to 7, {2, 3, 5, 7}, reduced modulo 6 are {2, 3, 5, 1}.  This contains the reduced residue system modulo 6, which is {1, 5}, and 7 is clearly the smallest such prime. - _Vladimir Shevelev_, Feb 19 2013
%o A038026 (PARI) a(n)={
%o A038026 my(todo=(1<<n)-1,r=2,q=2);
%o A038026 if(n==1, return(2));
%o A038026 for(a=0,n-1,
%o A038026 if(gcd(a,n)>1,todo=bitnegimply(todo,1<<a))
%o A038026 );
%o A038026 todo=bitnegimply(todo,1<<2);
%o A038026 forprime(p=3,default(primelimit),
%o A038026 r+=p-q;
%o A038026 r=r%n;
%o A038026 todo=bitnegimply(todo,1<<r);
%o A038026 if(!todo, return(p));
%o A038026 q=p;
%o A038026 );
%o A038026 error("Not enough precomputed primes")
%o A038026 }; \\ _Charles R Greathouse IV_, Feb 14 2011
%o A038026 (PARI) p(n,b)=while(!isprime(b), b+= n); b
%o A038026 a(n)=my(t=p(n,1));for(b=2,n-1,if(gcd(n,b)==1,t=max(t,p(n,b))));t \\ _Charles R Greathouse IV_, Sep 08 2012
%Y A038026 This sequence is a lower bound for the related sequence A085420.
%Y A038026 Cf. A038025.
%K A038026 nonn
%O A038026 1,1
%A A038026 _Christian G. Bower_