A073648 -id:A073648 - cp's OEIS frontend

A022004 Initial members of prime triples (p, p+2, p+6).

5, 11, 17, 41, 101, 107, 191, 227, 311, 347, 461, 641, 821, 857, 881, 1091, 1277, 1301, 1427, 1481, 1487, 1607, 1871, 1997, 2081, 2237, 2267, 2657, 2687, 3251, 3461, 3527, 3671, 3917, 4001, 4127, 4517, 4637, 4787, 4931, 4967, 5231, 5477

Offset: 1

Warut Roonguthai

Subsequence of A001359. - R. J. Mathar, Feb 10 2013
All terms are congruent to 5 (mod 6). - Matt C. Anderson, May 22 2015
Intersection of A001359 and A023201. - Zak Seidov, Mar 12 2016

Matt C. Anderson Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
T. Forbes and Norman Luhn, Prime k-tuplets
R. J. Mathar, Table of Prime Gap Constellations (2013,2024), 275 pages (no not print...)
Thomas R. Nicely, Enumeration of the prime triples (q,q+2,q+6) to 1e16.
P. Pollack, Analytic and Combinatorial Number Theory, Course Notes, p. 132, ex. 3.4.3. [Broken link?]
P. Pollack, Analytic and Combinatorial Number Theory, Course Notes, p. 132, ex. 3.4.3.
Maxie D. Schmidt, New Congruences and Finite Difference Equations for Generalized Factorial Functions, arXiv:1701.04741 [math.CO], 2017.
Eric Weisstein's World of Mathematics, Prime Triplet
Index entries for primes, gaps between

Cf. A073648, A098412, A372247 (subsequence).
Cf. A001359, A023201.
Subsequence of A007529.

[ p: p in PrimesUpTo(10000) | IsPrime(p+2) and IsPrime(p+6) ] // Vincenzo Librandi, Nov 19 2010

A022004 := proc(n)
    if n= 1 then
        5;
    else
        for a from procname(n-1)+2 by 2 do
            if isprime(a) and isprime(a+2) and isprime(a+6) then
                return a;
            end if;
        end do:
    end if;
end proc: # R. J. Mathar, Jul 11 2012

Select[Prime[Range[1000]], PrimeQ[#+2] && PrimeQ[#+6]&] (* Vladimir Joseph Stephan Orlovsky, Mar 30 2011 *)
Transpose[Select[Partition[Prime[Range[1000]],3,1],Differences[#]=={2,4}&]][[1]] (* Harvey P. Dale, Dec 24 2011 *)

is(n)=isprime(n)&&isprime(n+2)&&isprime(n+6) \\ Charles R Greathouse IV, Jul 01 2013

from sympy import primerange
def aupto(limit):
  p, q, alst = 2, 3, []
  for r in primerange(5, limit+7):
    if p+2 == q and p+6 == r: alst.append(p)
    p, q = q, r
  return alst
print(aupto(5477)) # Michael S. Branicky, May 11 2021

A098412 Greatest members p of prime triples (p-6, p-4, p).

Original entry on oeis.org

11, 17, 23, 47, 107, 113, 197, 233, 317, 353, 467, 647, 827, 863, 887, 1097, 1283, 1307, 1433, 1487, 1493, 1613, 1877, 2003, 2087, 2243, 2273, 2663, 2693, 3257, 3467, 3533, 3677, 3923, 4007, 4133, 4523, 4643, 4793, 4937, 4973, 5237, 5483, 5507, 5657, 6203

Offset: 1

Reinhard Zumkeller, Sep 07 2004

nonn

Subsequence of A046117; a(n) = A073648(n) + 4 = A022004(n) + 6.

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

Cf. A098413, A098415.

[p: p in PrimesUpTo(6500)|IsPrime(p) and IsPrime(p-6) and IsPrime(p-4)]; // Vincenzo Librandi, Dec 26 2010

K:=10^7: # to get all terms <= K.
for n from 1 by 2 to K do; if isprime(n-6) and isprime(n-4) and isprime(n) then print(n) else fi; od;  # Muniru A Asiru, Aug 06 2017

Select[Table[Prime[n], {n, 1000}], PrimeQ[# - 4] && PrimeQ[# - 6] &] (* Vladimir Joseph Stephan Orlovsky, Jun 19 2011 *)
Select[Partition[Prime[Range[1000]],3,1],Differences[#]=={2,4}&][[All,3]] (* Harvey P. Dale, Sep 23 2017 *)

A073649 Define the composite field of a prime q to be f(q) = (t,s) if p, q, r are three consecutive primes and q-p = t and r-q = s. This is a sequence of primes q with field (4,2).

Original entry on oeis.org

11, 17, 41, 71, 101, 107, 197, 227, 281, 311, 461, 617, 827, 857, 881, 1091, 1301, 1427, 1451, 1487, 1667, 1697, 1787, 1871, 1877, 1997, 2087, 2141, 2381, 2687, 2711, 2801, 3167, 3257, 3461, 3467, 3851, 4157, 4517, 4787, 5231, 5417, 5441, 5651, 5657

Offset: 1

Amarnath Murthy, Aug 09 2002

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A073648, A073650.
Cf. A098413, A098414.
Equals A022005 + 4.

Transpose[Select[Partition[Prime[Range[1200]],3,1],Differences[#] == {4,2}&]] [[2]] (* Harvey P. Dale, Jul 23 2011 *)

Corrected and extended by Benoit Cloitre, Aug 13 2002

A172454 Primes p such that (p, p+2, p+6, p+12) is a prime quadruple.

Original entry on oeis.org

5, 11, 17, 41, 101, 227, 347, 641, 1091, 1277, 1427, 1481, 1487, 1607, 2687, 3527, 3917, 4001, 4127, 4637, 4787, 4931, 8231, 9461, 10331, 11777, 12107, 13901, 14627, 16061, 19421, 20747, 21011, 21557, 22271, 23741, 25577, 26681, 26711, 27737

Offset: 1

Michel Lagneau, Feb 03 2010

nonn

The four primes do not have to be consecutive. - Harvey P. Dale, Jul 23 2011

			The first two terms correspond to the quadruples (5,7,11,17) and (11,13,17,23).

R. K. Guy, Unsolved Problems in Number Theory, E30.
P. A. MacMahon, The prime numbers of measurement on a scale, Proc. Camb. Phil. Soc. 21 (1923), 651-654; reprinted in Coll. Papers I, pp. 797-800.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
G. E. Andrews, MacMahon's prime numbers of measurement, Amer. Math. Monthly, 82 (1975), 922-923.
R. L. Graham and C. B. A. Peck, Problem E1910, Amer. Math. Monthly, 75 (1968), 80-81.
Eric Weisstein's World of Mathematics, Prime Triplet.

Cf. A073648, A098412.

for n from 1 by 2 to 110000 do; if isprime(n) and isprime(n+2) and isprime(n+6) and isprime(n+12) then print(n) else fi;od;

Select[Prime[Range[3100]],And@@PrimeQ[{#+2,#+6,#+12}]&] (* Harvey P. Dale, Jul 23 2011 *)

forprime(p=2,1e4,if(isprime(p+2)&&isprime(p+6)&&isprime(p+12), print1(p", "))) \\ Charles R Greathouse IV, Mar 04 2012

A073650 Define the composite field of a prime q to be f(q) = (t,s) if p, q, r are three consecutive primes and q-p = t and r-q = s. This is a sequence of primes q with field (2,6).

Original entry on oeis.org

31, 61, 73, 151, 271, 433, 571, 601, 1033, 1063, 1231, 1291, 1321, 1453, 1621, 2131, 2341, 2383, 2551, 2713, 2791, 3301, 3541, 4021, 4051, 4093, 4651, 4723, 5101, 5443, 5521, 5641, 5743, 5851, 6271, 6361, 6571, 6703, 6961, 7213, 8011, 9001, 9043, 9343

Offset: 1

Amarnath Murthy, Aug 09 2002

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A073648, A073649.

Transpose[Select[Partition[Prime[Range[1200]],3,1],Differences[#] == {2,6}&]][[2]] (* Harvey P. Dale, Jul 23 2011 *)

More terms from Benoit Cloitre, Aug 13 2002

A172456 Primes p such that (p, p+2, p+6, p+12, p+14, p+20) is a prime sextuple.

Original entry on oeis.org

17, 1277, 1607, 3527, 4637, 71327, 97367, 113147, 191447, 290657, 312197, 416387, 418337, 421697, 450797, 566537, 795647, 886967, 922067, 1090877, 1179317, 1300127, 1464257, 1632467, 1749257, 1866857, 1901357, 2073347, 2322107

Offset: 1

Michel Lagneau, Feb 03 2010

nonn

The last digit of each of these prime numbers is 7.
Subsequence of A078946.
The primes are always consecutive: The few ways of inserting other primes are: (p,p+2,p+4)... [impossible since one of these would be a multiple of 3]; (p,p+2,p+6),(p+8),(p+12),(p+14) [impossible since one of these would be a multiple of 5]; (p,p+2,p+6),(p+10) [impossible since one of these would be a multiple of 3]; (p,p+2,p+6),(p+12),(p+14),(p+16) [impossible since one of these would be a multiple of 3]; (p,p+2,p+6),(p+12),(p+14),(p+18) [impossible since one of these would be a multiple of 5]. - R. J. Mathar, Jun 15 2013

			The first two terms correspond to the sextuples (17,19,23,29,31,37) and (1277,1279,1283,1289,1291,1297).

R. K. Guy, Unsolved Problems in Number Theory, E30.

Amiram Eldar, Table of n, a(n) for n = 1..1000
G. E. Andrews, MacMahon's prime numbers of measurement, Amer. Math. Monthly, 82 (1975), 922-923.
T. Forbes, Prime k-tuplets
R. L. Graham and C. B. A. Peck, Problem E1910, Amer. Math. Monthly, 75 (1968), 80-81.
P. A. MacMahon, The prime numbers of measurement on a scale, Proc. Camb. Phil. Soc. 21 (1923), 651-654; reprinted in Coll. Papers I, pp. 797-800.
Eric Weisstein's World of Mathematics, Prime Triplet.

Initial members of prime quadruples (p, p+2, p+6, p+12): A172454.

Cf. A073648, A098412.

for n from 1 by 2 to 400000 do; if isprime(n) and isprime(n+2) and isprime(n+6) and isprime(n+12) and isprime(n + 14) and isprime(n+20) then print(n) else fi;od;

Select[Prime[Range[171000]],And@@PrimeQ[{#+2,#+6,#+12,#+14,#+20}]&] (* Harvey P. Dale, Jul 23 2011 *)
Select[Prime[Range[171000]],AllTrue[#+{2,6,12,14,20},PrimeQ]&] (* or *) Select[ Partition[Prime[Range[171000]],6,1],Differences[#]=={2,4,6,2,6}&][[All,1]] (* Harvey P. Dale, Sep 04 2022 *)

A073651 Define the composite field of a prime q to be f(q) = (t,s) if p, q, r are three consecutive primes and q-p = t and r-q = s. This is a sequence of primes q with field (6,2).

Original entry on oeis.org

29, 59, 137, 179, 239, 269, 569, 599, 659, 1019, 1229, 1289, 1607, 1619, 2339, 2549, 2969, 3329, 3539, 3767, 3917, 3929, 4019, 4217, 4259, 4649, 4799, 5009, 5279, 5477, 5849, 5867, 6269, 6359, 6569, 6659, 6869, 7127, 7457, 7487, 7547, 7589, 8087, 8429, 8837, 8969, 9419, 9629

Offset: 1

Amarnath Murthy, Aug 09 2002

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A073648, A073649, A073650.

Transpose[Select[Partition[Prime[Range[1200]],3,1],Differences[#]=={6,2}&]][[2]] (* Harvey P. Dale, Jul 23 2011 *)

Corrected and extended by Ryan Propper, Jul 10 2005

Corrected and extended by Harvey P. Dale, Jul 23 2011

A143726 Middle members of beastly cousin prime triples: primes p such that both p+666 and p-666 are prime.

Original entry on oeis.org

733, 773, 823, 857, 877, 947, 997, 1033, 1087, 1123, 1213, 1223, 1283, 1307, 1327, 1423, 1487, 1607, 1993, 2027, 2137, 2153, 2237, 2273, 2287, 2333, 2543, 2663, 2677, 2693, 2797, 2803, 2917, 3187, 3257, 3323, 3407, 3433, 3463, 3467, 3593, 3623, 3847

Offset: 1

Vladimir Joseph Stephan Orlovsky, Aug 30 2008

nonn

			733 - 666 = 67, 733 + 666 = 1399 and 67, 733, 1399 are all prime, so 733 is a term of the sequence. - _Felix Fröhlich_, May 26 2022

Felix Fröhlich, Table of n, a(n) for n = 1..10000

Cf. A007529 (p, p+2 or +4, p+6 prime), A023200 (p and p+4 prime), A046132 (p-4 and p prime), A073648 (p-2, p and p+4 prime).

lst={};b=666;Do[p=Prime[n];If[PrimeQ[p+b]&&PrimeQ[p-b],AppendTo[lst,p]],{n,5!+2,7!}];lst
Select[Prime[Range[122,600]],AllTrue[#+{666,-666},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Apr 08 2018 *)

forprime(p=1, 3900, if(ispseudoprime(p+666) && ispseudoprime(p-666), print1(p, ", "))) \\ Felix Fröhlich, May 26 2022

Name edited by Felix Fröhlich, May 26 2022

A163635 a(n) = 3*A022004(n) + 8.

Original entry on oeis.org

23, 41, 59, 131, 311, 329, 581, 689, 941, 1049, 1391, 1931, 2471, 2579, 2651, 3281, 3839, 3911, 4289, 4451, 4469, 4829, 5621, 5999, 6251, 6719, 6809, 7979, 8069, 9761, 10391, 10589, 11021, 11759, 12011, 12389, 13559, 13919, 14369, 14801

Offset: 1

Vincenzo Librandi, Aug 02 2009

Sum of the members of the n-th prime triple (p, p+2, p+6).

All terms are congruent to 5 (mod 18). See A242215. - Robert Bilinski, Sep 24 2019

			23 is in the sequence because 23 = 5+7+11 = 3*5+8.
41 is in the sequence because 41 = 11+13+17 = 3*11+8.

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

Cf. A022004, A073648, A098412.

Cf. A242215.

[(3*p+8): p in PrimesUpTo(1000)| IsPrime(p+6) and IsPrime(p+2)]; // Vincenzo Librandi, Jan 06 2018

8 + 3*Select[Prime[Range[1000]], PrimeQ[# + 2] && PrimeQ[# + 6] &] (* Vincenzo Librandi, Jan 04 2014 *)

is(n)=n%18==5 && isprime(n\3-2) && isprime(n\3) && isprime(n\3+4) \\ Charles R Greathouse IV, Jan 06 2018

a(n) = A022004(n) + (A022004(n)+2) + (A022004(n)+6);

a(n) = A022004(n) + A073648(n) + A098412(n).

Notation normalized by R. J. Mathar, Aug 07 2009

A244452 Primes p such that p^2-2 and p^2+4 are also prime (i.e., initial members of prime triples (p, p^2-2, p^2+4)).

Original entry on oeis.org

3, 5, 7, 13, 37, 47, 103, 233, 293, 313, 607, 677, 743, 1367, 1447, 2087, 2543, 3023, 3803, 3863, 4093, 4153, 4373, 4583, 4643, 4793, 4957, 5087, 5153, 5623, 5683, 5923, 6287, 7177, 7247, 7547, 7817, 8093, 8527, 9133, 9403

Offset: 1

Felix Fröhlich, Jun 28 2014

nonn

Intersection of A062326 and A062324.

			3 is in the sequence since it is the first member of the triple (3, 3^2-2, 3^2+4) and the resulting values in the triple (3, 7, 13) are all prime.

Felix Fröhlich, Table of n, a(n) for n = 1..242507 (all terms up to 10^9)

Cf. A022004, A073648, A098413.

Select[Prime[Range[1200]],AllTrue[#^2+{4,-2},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Apr 28 2018 *)

forprime(p=2, 10^4, if(isprime(p^2-2) && isprime(p^2+4), print1(p, ", ")))

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A022004 Initial members of prime triples (p, p+2, p+6).

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A098412 Greatest members p of prime triples (p-6, p-4, p).

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A073649 Define the composite field of a prime q to be f(q) = (t,s) if p, q, r are three consecutive primes and q-p = t and r-q = s. This is a sequence of primes q with field (4,2).

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A172454 Primes p such that (p, p+2, p+6, p+12) is a prime quadruple.

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A073650 Define the composite field of a prime q to be f(q) = (t,s) if p, q, r are three consecutive primes and q-p = t and r-q = s. This is a sequence of primes q with field (2,6).

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A172456 Primes p such that (p, p+2, p+6, p+12, p+14, p+20) is a prime sextuple.

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A073651 Define the composite field of a prime q to be f(q) = (t,s) if p, q, r are three consecutive primes and q-p = t and r-q = s. This is a sequence of primes q with field (6,2).

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A143726 Middle members of beastly cousin prime triples: primes p such that both p+666 and p-666 are prime.

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