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

1, 9, 15, 54, 271, 271, 2209, 11199, 13717, 13717, 34369, 136456, 172146, 1254203, 1254203, 4308948, 12762142, 21357253, 25813464, 25813464, 39500857, 39500857, 947438659, 947438659, 947438659, 5703167678, 5703167678, 16976360924, 57446769091, 57446769091, 57446769091

Offset: 1

Author

M. F. Hasler, Sep 02 2016

Keywords

• nonn

Comments

First term of runs of increasing length of consecutive integers in A270190. - M. F. Hasler, Sep 03 2016

Examples

			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.
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.
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.
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.
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.
		

Links

• J. K. Andersen, Consecutive Congruent Primes.

Crossrefs

Cf. A054679; A247816, A247967; A091177, A091178; A000720, A000040.

Subsequence of A270190 (after discarding 1 and duplicates of other terms).

Programs

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))

Formula

a(n) = A000720(A054679(n)).

Extensions

a(30)-a(31) from and name clarified by Jinyuan Wang, Feb 24 2020