A054678
n consecutive primes differ by a multiple of 4 starting at a(n).
Original entry on oeis.org
7, 89, 199, 463, 463, 463, 11593, 11593, 183091, 241603, 241603, 241603, 9177431, 9177431, 12270077, 105639091, 297779117, 297779117, 727334879, 1113443017, 1113443017, 1113443017, 1113443017, 79263248027, 84676452781, 113391385603
Offset: 2
More terms from Larry Reeves (larryr(AT)acm.org), Nov 09 2000
A276414
Index of the first prime which starts a run of n consecutive primes all congruent to each other mod 3 (or mod 6).
Original entry on oeis.org
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
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.
Subsequence of
A270190 (after discarding 1 and duplicates of other terms).
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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))
a(30)-a(31) from and name clarified by
Jinyuan Wang, Feb 24 2020
A054680
n consecutive primes differ by a multiple of 8 starting at a(n).
Original entry on oeis.org
89, 1823, 20809, 73133, 989647, 3250469, 9065867, 35677501, 101341613, 1383423311, 11312238283, 19201563659, 132932904029, 534956098463, 925195153703, 20151469541389, 20151469541389, 102573904861013
Offset: 2
More terms from Larry Reeves (larryr(AT)acm.org), Nov 09 2000
More terms from Larry Soule (lsoule(AT)gmail.com), Jun 11 2006
A259360
Initial prime in the least set of exactly n+1 consecutive primes with n gaps all multiples of 4.
Original entry on oeis.org
7, 89, 199, 883, 12401, 463, 36551, 11593, 183091, 766261, 3358169, 241603, 11739307, 9177431, 12270077, 105639091, 310523021, 297779117, 727334879, 5344989829, 1481666377, 2572421893, 1113443017, 79263248027, 84676452781
Offset: 1
a(6)=463 because the first set of 7 consecutive primes is {463,467,479,487,491,499,503} with 6 gaps {4,12,8,4,8,4} all multiples of 4 while the next prime after 503 is 509 and 509-503=6 is not a multiple of 4.
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back(p,n)=while(n,p=precprime(p-1); n--); p
v=vector(20); g=0; p=2; forprime(q=3,1e6, if((q-p)%4, if(g&&g<=#v&&v[g]==0, v[g]=back(p,g)); g=0, g++);p=q); v \\ Charles R Greathouse IV, Jul 14 2015
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