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A276414 Index of the first prime which starts a run of n consecutive primes all congruent to each other mod 3 (or mod 6).

%I A276414 #34 Feb 24 2020 07:19:53
%S A276414 1,9,15,54,271,271,2209,11199,13717,13717,34369,136456,172146,1254203,
%T A276414 1254203,4308948,12762142,21357253,25813464,25813464,39500857,
%U A276414 39500857,947438659,947438659,947438659,5703167678,5703167678,16976360924,57446769091,57446769091,57446769091
%N A276414 Index of the first prime which starts a run of n consecutive primes all congruent to each other mod 3 (or mod 6).
%C A276414 First term of runs of increasing length of consecutive integers in A270190. - _M. F. Hasler_, Sep 03 2016
%H A276414 J. K. Andersen, <a href="http://primerecords.dk/congruent-primes.htm">Consecutive Congruent Primes</a>.
%F A276414 a(n) = A000720(A054679(n)).
%e A276414 prime(9) = 23 starts the first run of 2 consecutive primes, {23, 29}, which are congruent to each other (mod 6). Therefore a(2) = 9.
%e A276414 prime(15) = 47 starts the first run of 3 consecutive primes, {47, 53, 59}, which are congruent to each other (mod 6). Therefore a(3) = 15.
%e A276414 prime(54) = 251 starts the first run of 4 consecutive primes, {251, 257, 263, 269}, which are congruent to each other (mod 6). Therefore a(4) = 54.
%e A276414 prime(271) = 1741 starts the first run of 5 consecutive primes, {1741, 1747, 1753, 1759, 1777}, which are congruent to each other (mod 6). Therefore a(5) = 271. This is the first case where the primes are of the form 3k+1.
%e A276414 prime(271) = 1741 also starts the first run of 6 consecutive primes, {1741, 1747, 1753, 1759, 1777, 1783}, which are all congruent to each other (mod 6). Therefore a(6) = 271, too.
%o A276414 (PARI) m=c=i=o=0;print1(1);forprime(p=1,,i++;(o-o=p)%3&&(!c||!c=0)&&next;c++>m||next;print1(",",i-m=c))
%Y A276414 Cf. A054679; A247816, A247967; A091177, A091178; A000720, A000040.
%Y A276414 Subsequence of A270190 (after discarding 1 and duplicates of other terms).
%K A276414 nonn
%O A276414 1,2
%A A276414 _M. F. Hasler_, Sep 02 2016
%E A276414 a(30)-a(31) from and name clarified by _Jinyuan Wang_, Feb 24 2020