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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).
4
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
1,2
COMMENTS
First term of runs of increasing length of consecutive integers in A270190. - M. F. Hasler, Sep 03 2016
FORMULA
a(n) = A000720(A054679(n)).
EXAMPLE
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.
PROG
(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))
CROSSREFS
Subsequence of A270190 (after discarding 1 and duplicates of other terms).
Sequence in context: A305219 A039314 A146584 * A175033 A043137 A043917
KEYWORD
nonn
AUTHOR
M. F. Hasler, Sep 02 2016
EXTENSIONS
a(30)-a(31) from and name clarified by Jinyuan Wang, Feb 24 2020
STATUS
approved