A049488 -id:A049488 - cp's OEIS frontend

A231608 Table whose n-th row consists of primes p such that p + 2n is also prime, read by antidiagonals.

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

Offset: 1

T. D. Noe, Nov 26 2013

			The following sequences are read by antidiagonals
{3, 5, 11, 17, 29, 41, 59, 71, 101, 107,...}
{3, 7, 13, 19, 37, 43, 67, 79, 97, 103,...}
{5, 7, 11, 13, 17, 23, 31, 37, 41, 47,...}
{3, 5, 11, 23, 29, 53, 59, 71, 89, 101,...}
{3, 7, 13, 19, 31, 37, 43, 61, 73, 79,...}
{5, 7, 11, 17, 19, 29, 31, 41, 47, 59,...}
{3, 5, 17, 23, 29, 47, 53, 59, 83, 89,...}
{3, 7, 13, 31, 37, 43, 67, 73, 97, 151,...}
{5, 11, 13, 19, 23, 29, 41, 43, 53, 61,...}
{3, 11, 17, 23, 41, 47, 53, 59, 83, 89,...}
...

T. D. Noe, Rows n = 1..100 of triangle, flattened

Cf. A001359, A023200, A023201, A023202, A023203.
Cf. A046133, A153417, A049488, A153418, A153419.
Cf. A020483 (numbers in first column).
Cf. A086505 (numbers on the diagonal).

A231608 := proc(n,k)
    local j,p ;
    j := 0 ;
    p := 2;
    while j < k do
        if isprime(p+2*n ) then
            j := j+1 ;
        end if;
        if j = k then
            return p;
        end if;
        p := nextprime(p) ;
    end do:
end proc:
for n from 1 to 10 do
    for k from 1 to 10 do
        printf("%3d ",A231608(n,k)) ;
    end do;
    printf("\n") ;
end do: # R. J. Mathar, Nov 19 2014

nn = 10; t = Table[Select[Range[100*nn], PrimeQ[#] && PrimeQ[# + 2*n] &, nn], {n, nn}]; Table[t[[n-j+1, j]], {n, nn}, {j, n}]

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).

A272815 Prime pairs of the form (p, p+16).

Original entry on oeis.org

3, 19, 7, 23, 13, 29, 31, 47, 37, 53, 43, 59, 67, 83, 73, 89, 97, 113, 151, 167, 157, 173, 163, 179, 181, 197, 211, 227, 223, 239, 241, 257, 277, 293, 331, 347, 337, 353, 367, 383, 373, 389, 433, 449, 463, 479, 487, 503, 541, 557, 547, 563, 571

Offset: 1

Vincenzo Librandi, May 07 2016

p and p+16 are not necessarily consecutive primes: (1831, 1847) is the first pair of consecutive primes that belongs to the sequence.

			The prime pairs are (3, 19), (7, 23), (13, 29) etc.

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

Cf. A000040, A049488.

Cf. prime pairs of the form (p, p+k): A077800 (k=2), A094343 (k=4), A156274 (k=6), A156320 (k=8), A140445 (k=10), A156323 (k=12), A140446 (k=14), this sequence (k=16), A156328 (k=18), A272816 (k=20), A140447 (k=22).

&cat [[p,p+16]: p in PrimesUpTo(1000) | IsPrime(p+16)];

Flatten[{#, # + 16}&/@Select[Prime[Range[200]], PrimeQ[# + 16] &]]

a(2n+1) = A049488(n+1).

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))

A222227 Numbers n such that n and n + 16 are prime and there is a power of two in the interval (n,n+16).

Original entry on oeis.org

3, 7, 13, 31, 241, 65521, 1048573, 2305843009213693951

Offset: 1

Brad Clardy, Feb 23 2013

nonn

It is a conjecture that this is a finite sequence. A search was conducted out to 2^1500.

Cf. A049488, A221211.

//Program finds primes separated by an even number (called gap) which
//have a power of two between them. Program starts with the smallest
//power of two above gap. Primes less than this starting point can be
//checked by inspection.
gap:=16;
start:=Ilog2(gap)+1;
for i:= start to 1000 do
    powerof2:=2^i;
    for k:=powerof2-gap+1 to powerof2-1 by 2 do
        if (IsPrime(k) and IsPrime(k+gap)) then k;
        end if;
    end for;
end for;

A106064 Primes p such that 1p + 16 and 16p + 1 are primes.

Original entry on oeis.org

7, 37, 97, 151, 163, 181, 331, 337, 487, 547, 571, 643, 727, 757, 967, 1033, 1087, 1093, 1303, 1423, 1471, 1567, 1831, 1987, 2083, 2113, 2221, 2251, 2281, 2671, 2683, 3121, 3187, 3607, 3847, 3931, 4111, 4201, 4447, 4663, 4993, 5023, 5791, 6073, 6343, 6553

Offset: 1

Zak Seidov, May 07 2005

nonn

Intersection of A049488 and A155943. - Michel Marcus, Jan 20 2018

[p: p in PrimesUpTo(10000)| IsPrime(p+16) and IsPrime(16*p+1)]; // Vincenzo Librandi, Nov 13 2010

Select[Prime[Range[220]], PrimeQ[16#+1]&&PrimeQ[1#+16]&]

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

A231608 Table whose n-th row consists of primes p such that p + 2n is also prime, read by antidiagonals.

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Maple

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

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A272815 Prime pairs of the form (p, p+16).

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Magma

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Formula

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

A222227 Numbers n such that n and n + 16 are prime and there is a power of two in the interval (n,n+16).

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Magma

A106064 Primes p such that 1*p + 16 and 16*p + 1 are primes.

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Magma

Mathematica

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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PARI

A106064 Primes p such that 1p + 16 and 16p + 1 are primes.