A331555 -id:A331555 - cp's OEIS frontend

A330366 Prime numbers p_k such that p_k == 1 (mod 10) and p_(k+1) == 1 (mod 10).

181, 241, 421, 631, 691, 811, 1021, 1051, 1171, 1471, 1801, 2521, 2731, 3001, 3361, 3571, 4201, 4231, 4261, 4831, 4861, 5011, 5351, 5581, 5701, 5791, 6091, 6121, 6301, 6481, 6491, 6691, 6781, 7321, 8101, 8221, 8821, 8941, 9421, 9511, 9931, 10141, 10321, 10771, 11161, 11971

Offset: 1

A.H.M. Smeets, Dec 12 2019

nonn

A.H.M. Smeets, Table of n, a(n) for n = 1..20000
R. J. Lemke Oliver and K. Soundararajan, Unexpected biases in the distribution of consecutive primes, arXiv:1603.03720 [math.NT], 2016.
R. J. Lemke Oliver and K. Soundararajan, Unexpected biases in the distribution of consecutive primes, Proceedings of the National Academy of Sciences of the United States of America, Vol. 113, No. 31 (2016), E4446-E4454.

Cf. A030430 (1, any), this sequence (1, 1), A331555 (1, 3), A331324 (1, 7), A030431 (3, any), A030432 (7, any), A030433 (9, any) [where (a, b) means p_k == a (mod 10) and p_(k+1) == b (mod 10)].

[p: p in PrimesUpTo(14000)| (p mod 10 eq 1) and (NextPrime(p) mod 10 eq 1)]; // Marius A. Burtea, Jan 20 2020

First @ Transpose @ Select[Partition[Select[Range[13500], PrimeQ], 2, 1], Mod[First[#], 10] == 1 && Mod[Last[#],10] == 1 &] (* Amiram Eldar, Jan 20 2020 *)

isok(p) = isprime(p) && ((p % 10)==1) && ((nextprime(p+1) % 10) == 1); \\ Michel Marcus, Jan 20 2020

A331324 Prime numbers p_k such that p_k == 1 (mod 10) and p_(k+1) == 7 (mod 10).

Original entry on oeis.org

31, 61, 131, 151, 251, 271, 331, 541, 571, 601, 751, 941, 971, 991, 1181, 1231, 1291, 1321, 1361, 1601, 1621, 1741, 1831, 1861, 1901, 2011, 2131, 2221, 2251, 2281, 2341, 2351, 2371, 2411, 2441, 2551, 2671, 2791, 2851, 3061, 3121, 3181, 3301, 3391, 3511, 3541, 3631, 3691, 3761, 3911

Offset: 1

A.H.M. Smeets, Jan 20 2020

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..1000
R. J. Lemke Oliver and K. Soundararajan, Unexpected biases in the distribution of consecutive primes, arXiv:1603.03720 [math.NT], 2016.
R. J. Lemke Oliver and K. Soundararajan, Unexpected biases in the distribution of consecutive primes, Proceedings of the National Academy of Sciences of the United States of America, Vol. 113, No. 31 (2016), E4446-E4454.

Cf. A030430 (1, any), A330366 (1, 1), A331555 (1, 3), this sequence (1, 7), A030431 (3, any), A030432 (7, any), A030433 (9, any) [where (a, b) means p_k == a (mod 10) and p_(k+1) == b (mod 10)].

[p: p in PrimesUpTo(4400)| (p mod 10 eq 1) and (NextPrime(p) mod 10 eq 7)]; // Marius A. Burtea, Jan 20 2020

First @ Transpose @ Select[Partition[Select[Range[4500], PrimeQ], 2, 1], Mod[First[#], 10] == 1 && Mod[Last[#],10] == 7 &] (* Amiram Eldar, Jan 20 2020 *)
Prime[#]&/@SequencePosition[Mod[Prime[Range[600]],10],{1,7}][[All,1]] (* Harvey P. Dale, Oct 17 2022 *)

A332674 Prime numbers p_k such that p_k == 1 (mod 10) and p_(k+1) == 9 (mod 10).

Original entry on oeis.org

401, 491, 701, 761, 911, 1381, 1571, 2161, 2531, 2741, 2861, 2971, 3011, 3041, 3221, 3271, 3491, 3701, 3881, 4751, 5051, 5171, 6011, 6221, 6451, 6521, 6581, 7151, 7351, 7621, 7691, 8171, 8191, 8681, 8761, 8971, 9311, 9941, 10151, 10391, 10531, 10631, 10691

Offset: 1

A.H.M. Smeets, Feb 19 2020

nonn

Robert Israel, Table of n, a(n) for n = 1..10000
R. J. Lemke Oliver and K. Soundararajan, Unexpected biases in the distribution of consecutive primes, arXiv:1603.03720 [math.NT], 2016.
R. J. Lemke Oliver and K. Soundararajan, Unexpected biases in the distribution of consecutive primes, Proceedings of the National Academy of Sciences of the United States of America, Vol. 113, No. 31 (2016), E4446-E4454.

Cf. A030430 (1, any), A330366 (1, 1), A331555 (1, 3), A331324 (1, 7), this sequence (1, 9), A030431 (3, any), A332675 (3, 1), A332676 (3, 3), A030432 (7, any), A030433 (9, any) [where (a, b) means p_k == a (mod 10) and p_(k+1) == b (mod 10)].

select(p -> isprime(p) and nextprime(p) mod 10 = 9, [seq(i,i=1..20000,10)]); # Robert Israel, Jun 10 2024

First @ Transpose @ Select[Partition[Select[Range[12500], PrimeQ], 2, 1], Mod[First[#], 10] == 1 && Mod[Last[#], 10] == 9 &] (* Amiram Eldar, Feb 19 2020 *)

forprime(p=1+o=2,1e4,p%10==9&&o%10==1&&print1(o",");o=p) \\ M. F. Hasler, Feb 19 2020

A332676 Prime numbers p_k such that p_k == 3 (mod 10) and p_(k+1) == 3 (mod 10).

Original entry on oeis.org

283, 1153, 1723, 2053, 2143, 3413, 3583, 3823, 3853, 4243, 4273, 4363, 4483, 4663, 5323, 5903, 6133, 6163, 6343, 6553, 6793, 6803, 7253, 7963, 8243, 8353, 8543, 8563, 8783, 8893, 9283, 9403, 10223, 10303, 10433, 10993, 11093, 11383, 12253, 12703, 13063, 13513, 13933, 14293, 14983

Offset: 1

A.H.M. Smeets, Feb 19 2020

nonn

Robert Israel, Table of n, a(n) for n = 1..10000
R. J. Lemke Oliver and K. Soundararajan, Unexpected biases in the distribution of consecutive primes, arXiv:1603.03720 [math.NT], 2016.
R. J. Lemke Oliver and K. Soundararajan, Unexpected biases in the distribution of consecutive primes, Proceedings of the National Academy of Sciences of the United States of America, Vol. 113, No. 31 (2016), E4446-E4454.

Cf. A030430 (1, any), A330366 (1, 1), A331555 (1, 3), A331324 (1, 7), A332674 (1, 9), A030431 (3, any), A332675 (3, 1), this sequence (3, 3), A030432 (7, any), A030433 (9, any) [where (a, b) means p_k == a (mod 10) and p_(k+1) == b (mod 10)].

filter:= t -> isprime(t) and nextprime(t) mod 10 = 3:
select(filter, [seq(i,i=3..20000,10)]); # Robert Israel, May 08 2020

First @ Transpose @ Select[Partition[Select[Range[20000], PrimeQ], 2, 1], Mod[First[#], 10] == 3 && Mod[Last[#], 10] == 3 &] (* Amiram Eldar, Feb 19 2020 *)

A332675 Prime numbers p_k such that p_k == 3 (mod 10) and p_(k+1) == 1 (mod 10).

Original entry on oeis.org

523, 683, 743, 983, 1163, 1193, 1373, 1523, 1733, 1823, 1913, 2003, 2153, 2213, 2243, 2273, 2503, 2663, 2843, 3623, 3803, 4373, 4423, 4463, 4583, 4603, 4703, 4733, 4943, 5483, 5573, 5693, 5783, 5813, 5953, 6113, 6143, 6203, 6473, 6833, 6983, 7393, 7433, 7673, 7883, 8093, 8513, 8573

Offset: 1

A.H.M. Smeets, Feb 19 2020

nonn

R. J. Lemke Oliver and K. Soundararajan, Unexpected biases in the distribution of consecutive primes, arXiv:1603.03720 [math.NT], 2016.
R. J. Lemke Oliver and K. Soundararajan, Unexpected biases in the distribution of consecutive primes, Proceedings of the National Academy of Sciences of the United States of America, Vol. 113, No. 31 (2016), E4446-E4454.

Cf. A030430 (1, any), A330366 (1, 1), A331555 (1, 3), A331324 (1, 7), A332674 (1, 9), A030431 (3, any), this sequence (3, 1), A332676 (3, 3), A030432 (7, any), A030433 (9, any) [where (a, b) means p_k == a (mod 10) and p_(k+1) == b (mod 10)].

First @ Transpose @ Select[Partition[Select[Range[10^4], PrimeQ], 2, 1], Mod[First[#], 10] == 3 && Mod[Last[#], 10] == 1 &] (* Amiram Eldar, Feb 19 2020 *)

cp's OEIS Frontend

A330366 Prime numbers p_k such that p_k == 1 (mod 10) and p_(k+1) == 1 (mod 10).

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A331324 Prime numbers p_k such that p_k == 1 (mod 10) and p_(k+1) == 7 (mod 10).

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A332674 Prime numbers p_k such that p_k == 1 (mod 10) and p_(k+1) == 9 (mod 10).

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A332676 Prime numbers p_k such that p_k == 3 (mod 10) and p_(k+1) == 3 (mod 10).

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Maple

Mathematica

A332675 Prime numbers p_k such that p_k == 3 (mod 10) and p_(k+1) == 1 (mod 10).

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Author

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Mathematica