A046133 -id:A046133 - cp's OEIS frontend

A156323 List of prime pairs of the form (p, p+12).

5, 17, 7, 19, 11, 23, 17, 29, 19, 31, 29, 41, 31, 43, 41, 53, 47, 59, 59, 71, 61, 73, 67, 79, 71, 83, 89, 101, 97, 109, 101, 113, 127, 139, 137, 149, 139, 151, 151, 163, 167, 179, 179, 191, 181, 193, 199, 211, 211, 223, 227, 239, 229, 241, 239, 251, 251, 263, 257

Offset: 1

Vincenzo Librandi, Feb 08 2009

			For p=5, 5+12=17, (5,17); p=59, 59+12=71, (59,71)

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

Cf. A046133, A092216.

Flatten[Select[{#,#+12}&/@Prime[Range[100]], PrimeQ[Last[#]]&]]  (* Harvey P. Dale, Mar 23 2011 *)

A164566 Primes p such that 7p-6 and 7p+6 are also prime numbers.

Original entry on oeis.org

5, 11, 19, 31, 41, 61, 71, 109, 151, 211, 229, 269, 379, 419, 431, 439, 479, 619, 641, 709, 739, 809, 839, 971, 1009, 1069, 1229, 1259, 1319, 1361, 1439, 1451, 1499, 1531, 1579, 1669, 1801, 1879, 1889, 2011, 2111, 2239, 2269, 2381, 2411, 2551, 2579, 2591

Offset: 1

Vladimir Joseph Stephan Orlovsky, Aug 16 2009

Primes of the form A087681(k)/7, any index k.

			For p=5, both 7*5-6=29 and 7*5+6=41 are prime,
for p=11, both 7*11-6=71 and 7*11+6=83 are prime.

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

Cf. A106015, A125272, A127430, A046133, A087681, A136052, A023225.

[p: p in PrimesUpTo(3000) | IsPrime(7*p-6) and IsPrime(7*p+6)]; // Vincenzo Librandi, Apr 09 2013

lst={};Do[p=Prime[n];If[PrimeQ[7*p-6]&&PrimeQ[7*p+6],AppendTo[lst,p]], {n,6!}];lst
Select[Prime[Range[700]], And @@ PrimeQ/@{7 # + 6, 7 # - 6}&] (* Vincenzo Librandi, Apr 09 2013 *)

is(n)=isprime(n) && isprime(7*n-6) && isprime(7*n+6) \\ Charles R Greathouse IV, Mar 28 2017

A136052 INTERSECT A023225. [R. J. Mathar, Aug 20 2009]

Examples rephrased by R. J. Mathar, Aug 20 2009

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}]

A046137 Primes p such that p+4 and p+12 are also prime.

Original entry on oeis.org

7, 19, 67, 97, 127, 229, 397, 487, 739, 757, 907, 1009, 1279, 1447, 1567, 1597, 1609, 1867, 1999, 2239, 2269, 2377, 2539, 2659, 2707, 3037, 3217, 3319, 3457, 3529, 3697, 3877, 3907, 3919, 4639, 4789, 4999, 5167, 5437, 5569, 5647, 5689, 5737

Offset: 1

Eric W. Weisstein

nonn

All terms == 1 (mod 6). - Robert Israel, Jul 24 2015

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime Triplet

select(t -> isprime(t) and isprime(t+4) and isprime(t+12), [6*i+1 $ i=1..1000]); # Robert Israel, Jul 24 2015

Select[Range@ 5760, AllTrue[{#, # + 4, # + 12}, PrimeQ] &] (* Michael De Vlieger, Jul 24 2015, Version 10 *)
Select[Prime[Range[800]],AllTrue[#+{4,12},PrimeQ]&] (* Harvey P. Dale, Apr 18 2022 *)

lista(nn) = forprime(p=2, nn, if (isprime(p+4) && isprime(p+12), print1(p, ", "))); \\ Michel Marcus, Jul 24 2015

A023200 INTERSECT A046133. - R. J. Mathar, Jan 23 2009

A046135 Primes p such that p+2 and p+12 are primes.

Original entry on oeis.org

5, 11, 17, 29, 41, 59, 71, 101, 137, 179, 227, 239, 269, 281, 347, 419, 431, 641, 809, 827, 1019, 1049, 1091, 1151, 1277, 1289, 1427, 1481, 1487, 1607, 1697, 1721, 1877, 2027, 2087, 2129, 2141, 2339, 2381, 2687, 2729, 2789, 2999, 3359, 3527, 3581

Offset: 1

Eric W. Weisstein

From Jonathan Vos Post, May 17 2006: (Start)
Could be defined as "Numbers n such that k^3+k^2+n is prime for k = 0, 1, 2."
The following subset is also prime for k = 3: 5, 11, 17, 71, 101, 137, 227, 281, 347, 431, 641, 827, 1151, 1277, 1487.  The following subset of those is also prime for k = 4: 17, 71, 101, 227, 827, 1151, 1487.  The following subset of those is also prime for k = 5: 827, 1151, 1487.  The "17" in A050266's n^3+n^2+17 is because k^3+k^2+17 is prime for k = 1,2,3,4,5,6,7,8,9,10. Between 10000 and 20000 there are 30 members of the k = 0,1,2 sequence, of which these 10 are also prime for k = 3: 10301, 10937, 11057, 11777, 12107, 13997, 15137, 15737, 16061, 19541.  The following subset of those is also prime for k = 5: 15137, 15737, 16061.  Somewhere in these sequences is a value that breaks the 11-term record of A050266 and indeed any known prime generating polynomial record. (End)

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Prime Triplet

Cf. A000040, A001359, A046133, A050266.

[p: p in PrimesUpTo(3600) | IsPrime(p+2) and IsPrime(p+12)]; // Vincenzo Librandi, Apr 09 2013

Select[Prime[Range[600]], PrimeQ[# + 2] && PrimeQ[# + 12]&] (* Vincenzo Librandi, Apr 09 2013 *)
Select[Prime[Range[600]],AllTrue[#+{2,12},PrimeQ]&] (* Harvey P. Dale, Jun 26 2025 *)

{n such that n prime, n+2 prime, n+12 prime} = A001359 INTERSECT A046133. - Jonathan Vos Post, May 17 2006

Edited by R. J. Mathar and N. J. A. Sloane, Aug 13 2008

A046141 p, p+8 and p+12 are primes.

Original entry on oeis.org

5, 11, 29, 59, 71, 89, 101, 269, 389, 431, 449, 479, 491, 761, 929, 1289, 1439, 1481, 1559, 1571, 1601, 2129, 2339, 2381, 2531, 2609, 2699, 2741, 2789, 3011, 3209, 3449, 3911, 4721, 5471, 5519, 5639, 5849, 6569, 6899, 6959

Offset: 1

Eric W. Weisstein

nonn

R. J. Mathar, Table of n, a(n) for n = 1..3766
Eric Weisstein's World of Mathematics, Prime Triplet.

Cf. A023202, A046133.

Select[Range@ 7000, AllTrue[{#, # + 8, # + 12}, PrimeQ] &] (* Michael De Vlieger, Jul 24 2015, Version 10 *)
Select[Prime[Range[1000]],AllTrue[#+{8,12},PrimeQ]&] (* Harvey P. Dale, Oct 07 2023 *)

lista(nn) = forprime(p=2, nn, if (isprime(p+8) && isprime(p+12), print1(p, ", "))); \\ Michel Marcus, Jul 24 2015

A086136 Primes p such that p and p+12 are nonconsecutive primes.

Original entry on oeis.org

5, 7, 11, 17, 19, 29, 31, 41, 47, 59, 61, 67, 71, 89, 97, 101, 127, 137, 139, 151, 167, 179, 181, 227, 229, 239, 251, 257, 269, 271, 281, 337, 347, 367, 389, 397, 409, 419, 421, 431, 449, 479, 487, 491, 557, 587, 601, 607, 631, 641, 647, 727, 739, 757, 761, 809

Offset: 1

Labos Elemer, Jul 28 2003

nonn

This sequence differs from A046133 because here the terms of A031930 are missing.

Complement of a=A031930 with respect to b=A046133: [b] & [not a]: this and A031930 are disjoint, but A031930 is a proper subset of A046133.

			First two deviations from A046133 are that 199=A031930(1) and 211=A031930(2) are missing. First 20 terms are equal.

Jonathan Frech, Table of n, a(n) for n = 1..1000

Cf. A031930, A046133.

Do[s=Prime[n]; s1=Prime[n+1]; If[PrimeQ[s+d]&& !Equal[s1-s, d], Print[s]], {n, 1, 1000}]; d=12
ncpQ[n_]:=Module[{c=n+12},PrimeQ[c]&&NextPrime[n]!=c]; Select[Prime[ Range[ 150]],ncpQ] (* Harvey P. Dale, May 03 2012 *)

A185022 Prime p such that p, p+12, p+24 are all primes.

Original entry on oeis.org

5, 7, 17, 19, 29, 47, 59, 89, 127, 139, 167, 199, 227, 239, 257, 269, 397, 409, 419, 467, 479, 607, 619, 727, 797, 929, 997, 1009, 1039, 1277, 1279, 1427, 1447, 1459, 1487, 1499, 1559, 1597, 1697, 1709, 1777, 1877, 1889, 1987, 2087, 2129, 2269, 2399, 2609

Offset: 1

Salvatore Di Guida, May 19 2012

Intersection of A046133 and A033560. - M. F. Hasler, May 19 2012

Salvatore Di Guida, Table of n, a(n) for n = 1..1000

Select[Range[50], PrimeQ[#] && PrimeQ[# + 12] && PrimeQ[# + 24] &] (* G. C. Greubel, Jun 20 2017 *)

forprime(p=1,2999,isprime(p+12)&isprime(p+24)&print1(p",")) \\ M. F. Hasler, May 19 2012

A056775 Numbers k such that phi(k+12) = phi(k) + 12.

Original entry on oeis.org

5, 7, 11, 17, 19, 29, 31, 41, 45, 47, 59, 61, 65, 67, 71, 80, 89, 97, 99, 101, 112, 117, 127, 135, 137, 139, 151, 167, 171, 176, 179, 181, 196, 199, 207, 209, 211, 227, 229, 239, 251, 257, 269, 271, 272, 279, 281, 294, 304, 310, 312, 337, 347, 367, 369, 389

Offset: 1

Labos Elemer, Aug 17 2000

nonn

Prime solutions are in A046133, common with primes in A015917.

			65 is a term since phi(65) = 48, phi(65+12) = phi(77) = 60 = 48 + 12.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Cf. A000010, A001838, A015917, A054902, A046133, A046133.

Select[Range[400],EulerPhi[#]+12==EulerPhi[#+12]&] (* Harvey P. Dale, Jan 21 2013 *)

A056776 Composite numbers k such that phi(k+12) = phi(k) + 12.

Original entry on oeis.org

45, 65, 80, 99, 112, 117, 135, 171, 176, 196, 207, 209, 272, 279, 294, 304, 310, 312, 369, 406, 429, 477, 496, 531, 592, 656, 657, 711, 752, 801, 909, 927, 944, 981, 1014, 1072, 1078, 1179, 1251, 1359, 1424, 1557, 1611, 1629, 1712, 1719, 1744, 1786, 1791

Offset: 1

Labos Elemer, Aug 17 2000

nonn

There are common cases with A054902.

			656 is a term since it is composite and phi(656) = 320, phi(656+12) = phi(668) = 332 = 320 + 12.
657 is a term since it is composite and phi(657) = 432, phi(657+12) = phi(669) = 444 = 432 + 12.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000010, A001838, A015917, A054902, A046133, A046133.

Select[Range[1800], CompositeQ[#] && EulerPhi[# + 12] == EulerPhi[#] + 12 &] (* Amiram Eldar, Mar 01 2020 *)

cp's OEIS Frontend

A156323 List of prime pairs of the form (p, p+12).

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A164566 Primes p such that 7*p-6 and 7*p+6 are also prime numbers.

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Magma

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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

Mathematica

A046137 Primes p such that p+4 and p+12 are also prime.

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Maple

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A046135 Primes p such that p+2 and p+12 are primes.

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Magma

Mathematica

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A046141 p, p+8 and p+12 are primes.

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PARI

A086136 Primes p such that p and p+12 are nonconsecutive primes.

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Mathematica

A185022 Prime p such that p, p+12, p+24 are all primes.

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A164566 Primes p such that 7p-6 and 7p+6 are also prime numbers.