A153417 -id:A153417 - cp's OEIS frontend

A153418 Primes p such that p+18 is also prime.

5, 11, 13, 19, 23, 29, 41, 43, 53, 61, 71, 79, 83, 89, 109, 113, 131, 139, 149, 163, 173, 179, 181, 193, 211, 223, 233, 239, 251, 263, 293, 313, 331, 349, 379, 383, 401, 421, 431, 439, 443, 449, 461, 491, 503, 523, 569, 599, 601, 613, 641, 643, 659, 673, 683

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 25 2008

Both p and p+18 have the same digital root (A010888). - Zak Seidov, Sep 14 2015
No term belongs to A030432. - Michel Marcus, Sep 14 2015
No term belongs to A045437. - Bruno Berselli, Sep 14 2015

			5 is in sequence because 5+18=23 is also prime;
11 is in sequence because 11+18=29 is also prime.

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

Cf. A007953, A010888, A045437, A049488, A030432, A153417.

A031936 is a subsequence. - Zak Seidov, Sep 13 2015

[p: p in PrimesUpTo(1000) | IsPrime(p+18)]; // Vincenzo Librandi, Apr 14 2013

lst={};d=18;Do[p=Prime[n];If[PrimeQ[p+d],AppendTo[lst,p]],{n,6!}];lst
Select[Prime[Range[150]], PrimeQ[(# + 18)]&] (* Vincenzo Librandi, Apr 14 2013 *)

list(n)=forprime(p=1,n,if(isprime(p+18),print1(p", ")))  \\ Anders Hellström, Sep 13 2015

[p for p in primes(700) if is_prime(p+18)]; # Bruno Berselli, Sep 14 2015

Definition improved by Bruno Berselli, Oct 31 2012

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

Original entry on oeis.org

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 *)

A153419 Primes p such that p+20 is also prime.

Original entry on oeis.org

3, 11, 17, 23, 41, 47, 53, 59, 83, 89, 107, 131, 137, 173, 179, 191, 251, 257, 263, 293, 311, 317, 347, 353, 359, 389, 401, 419, 443, 467, 479, 503, 521, 557, 587, 593, 599, 641, 653, 719, 809, 839, 857, 863, 887, 947, 971, 977, 1013, 1019, 1031, 1049, 1097

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 25 2008

			3 is in the sequence because 3+20=23 is prime; 11 is in the sequence because 11+20=31 is prime.

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (first 1000 terms from Vincenzo Librandi)

Cf. A153417, A049488, A153418.

[p: p in PrimesUpTo(1100) | IsPrime(p + 20)]; // Vincenzo Librandi, Apr 14 2013

for a from 1 to 140 do if isprime(a) and isprime(a+20) then print(a)
  end if;  end do; # Matt C. Anderson, Jun 20 2022

Select[Prime[Range[200]], PrimeQ[(# + 20)]&] (* Vincenzo Librandi, Apr 14 2013 *)

Definition improved from Bruno Berselli, 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] &]

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

Original entry on oeis.org

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}]

A282423 a(n) = smallest k such that A282026(k) = n, or 0 if no such k exists.

Original entry on oeis.org

3, 2, 0, 13, 19, 0, 427, 4, 0, 0, 1, 0, 802, 99412, 0, 3097, 7, 0, 637, 0, 0, 7225627, 30898822, 0, 0, 280134277, 0, 31705902442, 43190647, 0, 965577112

Offset: 1

Andrey Zabolotskiy and Altug Alkan, Feb 14 2017, following a suggestion from N. J. A. Sloane

a(n) is nonzero if n is in A282429.

For n>4 and nonzero a(n), 2*a(n)+3 is in A022004. For n>8 and nonzero a(n), 2*a(n)+3 is also in A153417. For n>16 and nonzero a(n), 2*a(n)+3 is also in A049481.

			a(10) = 0. Proof: Suppose 10 is a term of A282026. For the corresponding n, 2*n + 1 cannot be divisible by 5 because of A282026’s definition (gcd(10, 2*n + 1) = 1). So 2*n + 1 can be only of the form 10*k + 1, 10*k + 3, 10*k + 7, 10*k + 9. But 10*k + 1 + 2*2, 10*k + 3 + 2*1, 10*k + 7 + 2*4, 10*k + 9 + 2*8 are all composite and 1, 2, 4, 8 are relatively prime to any odd number. Since all of them are smaller than 10, this is the contradiction to the assumption that 10 is the term which is the smallest number for corresponding n. This also proves that a(5*k) = 0 for any k > 1.

Cf. A282026, A282429.

A161866 Numbers k such that k^2+k+7 and k^2+k-7 are both prime.

Original entry on oeis.org

3, 5, 9, 12, 24, 29, 32, 39, 44, 50, 57, 59, 65, 102, 135, 137, 144, 170, 180, 207, 260, 267, 297, 302, 305, 344, 347, 360, 365, 369, 389, 404, 429, 464, 474, 495, 540, 555, 570, 612, 620, 659, 662, 689, 767, 774, 792, 824, 837, 872, 885, 900, 950, 954, 989

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jun 20 2009

			a(1)=3 as 12+-7 are primes. a(2)=5 as 30+-7 are primes.

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Cf. A088485, A161863, A161864, A153417.

[k:k in [1..1000]| IsPrime(k^2+k+7) and IsPrime(k^2+k-7)]; // Marius A. Burtea, Feb 17 2020

q=7;lst7={};Do[p=n^2+n;If[PrimeQ[p-q]&&PrimeQ[p+q],AppendTo[lst7,n]], {n,0,7!}];lst7
Select[Range[1000],AllTrue[#^2+#+{7,-7},PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 26 2021 *)

Definition rephrased by R. J. Mathar, Jun 23 2009

A222424 Numbers n such that n and n + 14 are prime and there is a power of two in the interval (n,n+14).

Original entry on oeis.org

3, 5, 23, 29, 53, 59, 509, 1019, 2039, 262133, 262139, 536870909, 73786976294838206459, 2787593149816327892691964784081045188247543

Offset: 1

Brad Clardy, Feb 24 2013

nonn

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

Cf. A153417, 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:=14;
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;

A329262 Prime pairs of the form (30k - 7, 30k + 7).

Original entry on oeis.org

23, 37, 53, 67, 83, 97, 113, 127, 263, 277, 293, 307, 353, 367, 383, 397, 443, 457, 563, 577, 593, 607, 743, 757, 773, 787, 863, 877, 953, 967, 983, 997, 1103, 1117, 1223, 1237, 1283, 1297, 1433, 1447, 1553, 1567, 1583, 1597, 1613, 1627

Offset: 1

Harry E. Neel, Nov 09 2019

The terms of this sequence are created by pairing the terms of the primes when the terms 30k - 7 and 30k + 7 are both prime. As has been pointed out, it is only a guess as to whether, or not, that (like the twin primes, etc.) there is or is not an infinite number of these pairings.

			As 4 * 30 - 7 = 113 and 4 * 30 + 7 = 127 are prime, both 113 and 127 are in the sequence. - _David A. Corneth_, Nov 10 2019

Cf. A140446, A153417, A007510.

Odd- (resp. even-) indexed terms are a subsequence of A132235 (resp. A132231).

&cat[[30*k-7] cat [30*k+7]:k in [1..60]|IsPrime(30*k-7) and IsPrime(30*k+7)]; // Marius A. Burtea, Nov 17 2019

Select[Prime[Range[1000]], MemberQ[{7, 23}, Mod[#, 30]] &] (* Jinyuan Wang, Nov 16 2019 *)
Flatten[Select[Table[30n + {-7, 7}, {n, 90}], PrimeQ[#[[1]]] && PrimeQ[#[[2]]] &]] (* Alonso del Arte, Dec 07 2019 *)

first(n) = n+=(n%2); my(res=List(),todo=n); for(i=1,oo, if(isprime(30*i-7) && isprime(30*i+7), listput(res,30*i-7); listput(res,30*i+7); todo-=2; if(todo<=0, return(res)))) \\ David A. Corneth, Nov 10 2019

A106063 Primes p such that 1p + 14 and 14p + 1 are primes.

Original entry on oeis.org

3, 5, 17, 47, 53, 59, 83, 113, 149, 167, 269, 353, 419, 443, 449, 509, 563, 587, 599, 647, 659, 677, 719, 797, 977, 983, 1103, 1109, 1187, 1223, 1289, 1367, 1409, 1433, 1439, 2039, 2099, 2237, 2297, 2357, 2423, 2543, 2579, 2657, 2753, 2789, 2837, 2957, 3023

Offset: 1

Zak Seidov, May 07 2005

nonn

Sunsequence of A153417. - Michel Marcus, Jan 20 2018

[p: p in PrimesUpTo(10000)| IsPrime(p+14) and IsPrime(14*p+1)]; // Vincenzo Librandi, Nov 13 2010

Select[Prime[Range[220]], PrimeQ[14#+1]&&PrimeQ[1#+14]&]

cp's OEIS Frontend

A153418 Primes p such that p+18 is also prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Extensions

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

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

A153419 Primes p such that p+20 is also prime.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Extensions

A252089 Primes p such that p + 26 is prime.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A282423 a(n) = smallest k such that A282026(k) = n, or 0 if no such k exists.

Views

Author

Keywords

Comments

Examples

Crossrefs

A161866 Numbers k such that k^2+k+7 and k^2+k-7 are both prime.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Extensions

A222424 Numbers n such that n and n + 14 are prime and there is a power of two in the interval (n,n+14).

Views

Author

Keywords

Comments

Crossrefs

A106063 Primes p such that 1p + 14 and 14p + 1 are primes.