A049489 -id:A049489 - cp's OEIS frontend

A156104 Primes p such that p+36 is also prime.

5, 7, 11, 17, 23, 31, 37, 43, 47, 53, 61, 67, 71, 73, 101, 103, 113, 127, 131, 137, 157, 163, 191, 193, 197, 227, 233, 241, 257, 271, 277, 281, 311, 313, 317, 331, 337, 347, 353, 373, 383, 397, 421, 431, 443, 463, 467, 487, 521, 541, 557, 563, 571, 577, 607

Offset: 1

Vincenzo Librandi, Feb 08 2009

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A156112.

Cf. sequences of the type p+n are primes: A001359 (n=2), A023200 (n=4), A023201 (n=6), A023202 (n=8), A023203 (n=10), A046133 (n=12), A153417 (n=14), A049488 (n=16), A153418 (n=18), A153419 (n=20), A242476 (n=22), A033560 (n=24), A252089 (n=26), A252090 (n=28), A049481 (n=30), A049489 (n=32), A252091 (n=34), this sequence (n=36); A062284 (n=50), A049490 (n=64), A156105 (n=72), A156107 (n=144).

[p: p in PrimesUpTo(1000) | IsPrime(p + 36)]; // Vincenzo Librandi, Oct 31 2012

Select[Prime[Range[1000]], PrimeQ[(#+ 36)]&] (* Vincenzo Librandi, Oct 31 2012 *)

A252089 Primes p such that p + 26 is prime.

Original entry on oeis.org

3, 5, 11, 17, 41, 47, 53, 71, 83, 101, 113, 131, 137, 167, 173, 197, 251, 257, 281, 311, 347, 353, 383, 431, 461, 521, 587, 593, 617, 647, 683, 701, 743, 761, 797, 827, 857, 881, 911, 941, 971, 983, 1013, 1061, 1091, 1097, 1103, 1187, 1223, 1277, 1301, 1373

Offset: 1

Vincenzo Librandi, Dec 14 2014

			17 is in this sequence because 17+26 = 43 is prime.
431 is in this sequence because 431+26 = 457 is prime.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. sequences of the type p+n are primes: A001359 (n=2), A023200 (n=4), A023201 (n=6), A023202 (n=8), A023203 (n=10), A046133 (n=12), A153417 (n=14), A049488 (n=16), A153418 (n=18), A153419 (n=20), A242476 (n=22), A033560 (n=24), this sequence (n=26), A252090 (n=28), A049481 (n=30), A049489 (n=32), A252091 (n=34), A156104 (n=36); A062284 (n=50), A049490 (n=64), A156105 (n=72), A156107 (n=144).

[NthPrime(n): n in [1..250] | IsPrime(NthPrime(n)+26)];

Select[Prime[Range[200]], PrimeQ[# + 26] &]

A361485 Primes p such that p + 1024 is also prime.

Original entry on oeis.org

7, 37, 67, 73, 79, 127, 139, 157, 163, 193, 199, 277, 283, 337, 349, 409, 457, 463, 487, 499, 547, 577, 613, 643, 673, 709, 787, 823, 853, 877, 883, 907, 1039, 1063, 1087, 1117, 1129, 1213, 1249, 1327, 1399, 1423, 1453, 1567, 1597, 1609, 1663, 1669, 1753, 1777, 1873, 1879

Offset: 1

Elmo R. Oliveira, Mar 13 2023

All terms are == 1 (mod 6).

			139 and 139 + 1024 = 1163 are both prime.

Winston de Greef, Table of n, a(n) for n = 1..10000

Cf. A000040.

Cf. sequences of the type p + k are primes: A001359 (k = 2), A023200 (k = 4), A023202 (k = 8), A049488 (k = 16), A049489 (k = 32), A049490 (k = 64), A049491 (k = 128), A361483 (k = 256), A361484 (k = 512), this sequence (k = 1024).

lista(nn)=my(v=vector(nn), p=2); for(n=1, nn, until(isprime(p+1024), p=nextprime(p+1)); v[n]=p); v \\ Winston de Greef, Mar 20 2023

A361483 Primes p such that p + 256 is also prime.

Original entry on oeis.org

7, 13, 37, 61, 97, 103, 127, 163, 193, 211, 223, 307, 313, 331, 337, 397, 421, 463, 487, 541, 571, 601, 607, 631, 673, 691, 727, 757, 853, 907, 937, 967, 1021, 1033, 1051, 1063, 1117, 1153, 1171, 1231, 1237, 1297, 1303, 1327, 1381, 1453, 1531, 1567, 1621, 1657, 1693, 1723

Offset: 1

Elmo R. Oliveira, Mar 13 2023

All terms are == 1 (mod 6).

			61 and 61 + 256 = 317 are both prime.

Cf. A000040.

Cf. sequences of the type p + k are primes: A001359 (k = 2), A023200 (k = 4), A023202 (k = 8), A049488 (k = 16), A049489 (k = 32), A049490 (k = 64), A049491 (k = 128), this sequence (k = 256), A361484 (k = 512), A361485 (k = 1024).

A361484 Primes p such that p + 512 is also prime.

Original entry on oeis.org

11, 29, 59, 89, 101, 107, 131, 149, 179, 197, 227, 239, 257, 311, 317, 347, 479, 509, 521, 557, 617, 641, 659, 701, 719, 809, 887, 911, 941, 947, 971, 977, 1019, 1031, 1097, 1109, 1151, 1181, 1187, 1229, 1277, 1289, 1319, 1361, 1367, 1439, 1481, 1487, 1499, 1571, 1601

Offset: 1

Elmo R. Oliveira, Mar 13 2023

All terms are == 5 (mod 6).

			59 and 59 + 512 = 571 are both prime.

Cf. A000040.

Cf. sequences of the type p + k are primes: A001359 (k = 2), A023200 (k = 4), A023202 (k = 8), A049488 (k = 16), A049489 (k = 32), A049490 (k = 64), A049491 (k = 128), A361483 (k = 256), this sequence (k = 512), A361485 (k = 1024).

A361679 A(n,k) is the n-th prime p such that p + 2^k is also prime; square array A(n,k), n>=1, k>=1, read by antidiagonals.

Original entry on oeis.org

3, 3, 5, 3, 7, 11, 3, 5, 13, 17, 5, 7, 11, 19, 29, 3, 11, 13, 23, 37, 41, 3, 7, 29, 31, 29, 43, 59, 7, 11, 19, 41, 37, 53, 67, 71, 11, 13, 23, 37, 47, 43, 59, 79, 101, 7, 29, 37, 29, 43, 71, 67, 71, 97, 107, 5, 37, 59, 61, 53, 67, 107, 73, 89, 103, 137

Offset: 1

Alois P. Heinz, Mar 20 2023

			Square array A(n,k) begins:
    3,   3,   3,   3,   5,   3,   3,   7,  11,   7, ...
    5,   7,   5,   7,  11,   7,  11,  13,  29,  37, ...
   11,  13,  11,  13,  29,  19,  23,  37,  59,  67, ...
   17,  19,  23,  31,  41,  37,  29,  61,  89,  73, ...
   29,  37,  29,  37,  47,  43,  53,  97, 101,  79, ...
   41,  43,  53,  43,  71,  67,  71, 103, 107, 127, ...
   59,  67,  59,  67, 107,  73,  83, 127, 131, 139, ...
   71,  79,  71,  73, 131, 103, 101, 163, 149, 157, ...
  101,  97,  89,  97, 149, 109, 113, 193, 179, 163, ...
  107, 103, 101, 151, 167, 127, 149, 211, 197, 193, ...

Alois P. Heinz, Antidiagonals n = 1..200, flattened

Columns k=1-10 give: A001359, A023200, A023202, A049488, A049489, A049490, A049491, A361483, A361484, A361485.
Row n=1 gives A056206.
Main diagonal gives A361680.
Cf. A000040.

A:= proc() option remember; local f; f:= proc() [] end;
      proc(n, k) option remember; local p;
        p:= `if`(nops(f(k))=0, 1, f(k)[-1]);
        while nops(f(k))

A086149 Primes p such that p + 32 is also prime and there are seven primes between p and p + 32.

Original entry on oeis.org

29, 41, 416387, 626597, 6560987, 6937937, 25658429, 25658441, 29597411, 49136357, 51448361, 57405419, 90279461, 128469149, 137943341, 162189089, 165531251, 175182587, 227779679, 261672779, 290973491, 294536141, 311725847, 334388279, 422005247, 442768631, 503804657

Offset: 1

Labos Elemer, Jul 29 2003

nonn

Amiram Eldar, Table of n, a(n) for n = 1..1000

Subsequence of A049489.

cp[x_, y_] := Count[Table[PrimeQ[i], {i, x, y}], True] Do[s=Prime[n]; s1=Prime[n+1]; If[PrimeQ[s+32]&& Equal[cp[s+1, s+d-1], 7], Print[s]], {n, 1, 1000000}]
Select[Partition[Prime[Range[98*10^5]],9,1],Last[#]-First[#]==32&] [[All,1]] (* Harvey P. Dale, Sep 12 2017 *)

lista(pmax) = {my(ps = primes(9)); forprime(p = prime(10), pmax, ps = concat(vecextract(ps, "^1"), p); if(ps[9] - ps[1] == 32, print1(ps[1], ", ")));} \\ Amiram Eldar, Aug 15 2024

Definition clarified by Harvey P. Dale, Sep 12 2017

a(19)-a(27) from Amiram Eldar, Aug 15 2024

A270754 Numbers n such that n - 31, n - 1, n + 1 and n + 31 are consecutive primes.

Original entry on oeis.org

90438, 258918, 293862, 385740, 426162, 532950, 1073952, 1317192, 1318410, 1401318, 1565382, 1894338, 1986168, 2174772, 2612790, 2764788, 3390900, 3450048, 3618960, 3797250, 3961722, 3973062, 4074870, 4306230, 4648068, 4917360, 5351010, 5460492

Offset: 1

Karl V. Keller, Jr., Mar 22 2016

nonn

This sequence is a subsequence of A014574 (average of twin prime pairs) and A256753.
The terms ending in 0 are divisible by 30 (cf. A249674).
The terms ending in 2 and 8 are congruent to 12 mod 30 and 18 mod 30 respectively.
The numbers n - 31 and n + 1 belong to A049481 (p and p + 30 are primes) and A124596 (p where p + 30 is the next prime).
The numbers n - 31 and n - 1 belong to A049489 (p and p + 32 are primes).

			90438 is the average of the four consecutive primes 90407, 90437, 90439, 90469.
258918 is the average of the four consecutive primes 258887, 258917, 258919, 258949.

Karl V. Keller, Jr., Table of n, a(n) for n = 1..100000
Eric Weisstein's World of Mathematics, Twin Primes

Cf. A014574, A077800 (twin primes), A249674, A256753.

from sympy import isprime,prevprime,nextprime
for i in range(0,1000001,6):
   if isprime(i-1) and isprime(i+1) and prevprime(i-1) == i-31 and nextprime(i+1) == i+31 :  print (i,end=', ')

A309392 Square array read by downward antidiagonals: A(n, k) is the k-th prime p such that p + 2*n is also prime, or 0 if that prime does not exist.

Original entry on oeis.org

3, 5, 3, 11, 7, 5, 17, 13, 7, 3, 29, 19, 11, 5, 3, 41, 37, 13, 11, 7, 5, 59, 43, 17, 23, 13, 7, 3, 71, 67, 23, 29, 19, 11, 5, 3, 101, 79, 31, 53, 31, 17, 17, 7, 5, 107, 97, 37, 59, 37, 19, 23, 13, 11, 3, 137, 103, 41, 71, 43, 29, 29, 31, 13, 11, 7, 149, 109

Offset: 1

Felix Fröhlich, Jul 28 2019

The same as A231608 except that A231608 gives the upward antidiagonals of the array, while this sequence gives the downward antidiagonals.
Conjecture: All values are nonzero, i.e., for any even integer e there are infinitely many primes p such that p + e is also prime.
The conjecture is true if Polignac's conjecture is true.

			The array starts as follows:
3,  5, 11, 17, 29, 41, 59,  71, 101, 107, 137, 149, 179, 191
3,  7, 13, 19, 37, 43, 67,  79,  97, 103, 109, 127, 163, 193
5,  7, 11, 13, 17, 23, 31,  37,  41,  47,  53,  61,  67,  73
3,  5, 11, 23, 29, 53, 59,  71,  89, 101, 131, 149, 173, 191
3,  7, 13, 19, 31, 37, 43,  61,  73,  79,  97, 103, 127, 139
5,  7, 11, 17, 19, 29, 31,  41,  47,  59,  61,  67,  71,  89
3,  5, 17, 23, 29, 47, 53,  59,  83,  89, 113, 137, 149, 167
3,  7, 13, 31, 37, 43, 67,  73,  97, 151, 157, 163, 181, 211
5, 11, 13, 19, 23, 29, 41,  43,  53,  61,  71,  79,  83,  89
3, 11, 17, 23, 41, 47, 53,  59,  83,  89, 107, 131, 137, 173
7, 19, 31, 37, 61, 67, 79, 109, 127, 151, 157, 211, 229, 241
5,  7, 13, 17, 19, 23, 29,  37,  43,  47,  59,  73,  79,  83

Wikipedia, Polignac's conjecture

Cf. A231608.

Cf. A001359 (row 1), A023200 (row 2), A023201 (row 3), A023202 (row 4), A023203 (row 5), A046133 (row 6), A153417 (row 7), A049488 (row 8), A153418 (row 9), A153419 (row 10), A242476 (row 11), A033560 (row 12), A252089 (row 13), A252090 (row 14), A049481 (row 15), A049489 (row 16), A252091 (row 17), A156104 (row 18), A271347 (row 19), A271981 (row 20), A271982 (row 21), A272176 (row 22), A062284 (row 25), A049490 (row 32), A020483 (column 1).

row(n, terms) = my(i=0); forprime(p=1, , if(i>=terms, break); if(ispseudoprime(p+2*n), print1(p, ", "); i++))
array(rows, cols) = for(x=1, rows, row(x, cols); print(""))
array(12, 14) \\ Print initial 12 rows and 14 columns of the array

cp's OEIS Frontend

A156104 Primes p such that p+36 is also prime.

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Magma

Mathematica

A252089 Primes p such that p + 26 is prime.

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Magma

Mathematica

A361485 Primes p such that p + 1024 is also prime.

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PARI

A361483 Primes p such that p + 256 is also prime.

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A361484 Primes p such that p + 512 is also prime.

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A361679 A(n,k) is the n-th prime p such that p + 2^k is also prime; square array A(n,k), n>=1, k>=1, read by antidiagonals.

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Maple

A086149 Primes p such that p + 32 is also prime and there are seven primes between p and p + 32.

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Mathematica

PARI

Extensions

A270754 Numbers n such that n - 31, n - 1, n + 1 and n + 31 are consecutive primes.

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Python

A309392 Square array read by downward antidiagonals: A(n, k) is the k-th prime p such that p + 2*n is also prime, or 0 if that prime does not exist.

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