|
%I #14 Mar 22 2013 13:05:30
%S 1,4,10,24,76,102,196,74,104,348,314,345,86,660,443,1494,914,1329,
%T 2613,1635,1316,1856,1688,2589,2628,6423,3116,2165,6320,4445,7278,
%U 4743,16539,17783,6084,3806,6281,8946,15129,6266,10976,19538,16794,31160,32916,57041
%N Let p = A002145(n) be the n-th prime of the form 4k+3, then a(n) is the smallest number such that p is the smallest prime of the form 4k+3 for which 4*a(n)+2-p is prime.
%C It is conjectured that a(n) is defined for all positive integers.
%C This is also the index of first occurrence of the n-th prime in the form of 4k+3 in A214834.
%H Lei Zhou, <a href="/A217696/b217696.txt">Table of n, a(n) for n = 1..170</a>
%e n=1: the first prime in the form of 4k+3 is 3, 3+3=6=4*1+2, so a(1)=1;
%e n=2: the second prime in the form of 4k+3 is 7, 7+7=14=3+11=4*3+2, and 11 is also a prime in the form of 4k+3, so a(2)!=3. 7+11=18=4*4+2=3+15, and 15 is not a prime number. So a(2)=4.
%t goal = 46; plst = {}; pct = 0; clst = {}; n = -1; While[pct < goal,
%t n = n + 4; If[PrimeQ[n], AppendTo[plst, n]; AppendTo[clst, 0];
%t pct++]]; n = 2; cct = 0; While[cct < goal, n = n + 4; p1 = n + 1;
%t While[p1 = p1 - 4; p2 = n - p1; ! ((PrimeQ[p1]) && (PrimeQ[p2]) && (Mod[p2, 4] == 3))]; If[MemberQ[plst, p2], If[id = Position[plst, p2][[1, 1]]; clst[[id]] == 0, clst[[id]] = (n - 2)/4; cct++]]]; clst
%o (PARI) ok(n,p)=if(!isprime(n-p),return(0));forprime(q=2,p-1,if(q%4==3 && isprime(n-q),return(0)));1
%o a(n)=my(p,k); forprime(q=2,,if(q%4==3&&n--==0,p=q;break)); k=(p+1)/4; while(!ok(4*k+2,p),k++); k \\ _Charles R Greathouse IV_, Mar 19 2013
%Y Cf. A214834, A016825, A000040, A002145, A155642.
%K nonn
%O 1,2
%A _Lei Zhou_, Mar 19 2013
|
