A098412 -id:A098412 - 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

A073648 Middle members of prime triples {p, p+2, p+6}.

Original entry on oeis.org

7, 13, 19, 43, 103, 109, 193, 229, 313, 349, 463, 643, 823, 859, 883, 1093, 1279, 1303, 1429, 1483, 1489, 1609, 1873, 1999, 2083, 2239, 2269, 2659, 2689, 3253, 3463, 3529, 3673, 3919, 4003, 4129, 4519, 4639, 4789, 4933, 4969, 5233, 5479, 5503, 5653, 6199

Offset: 1

Amarnath Murthy, Aug 09 2002

Zak Seidov, Table of n,a(n) for n=1,2380, a(n)<2*10^6

Cf. A073649, A073650.

Cf. A098412, A098414.

Transpose[Select[Partition[Prime[Range[850]],3,1],Differences[#]=={2,4}&]][[2]]  (* Harvey P. Dale, Feb 20 2011 *)

a(n) = A022004(n) + 2.

More terms from Benoit Cloitre, Aug 13 2002

A098415 Greatest members r of prime triples (p,q,r) with p

Original entry on oeis.org

11, 13, 17, 19, 23, 43, 47, 73, 103, 107, 109, 113, 197, 199, 229, 233, 283, 313, 317, 353, 463, 467, 619, 647, 827, 829, 859, 863, 883, 887, 1093, 1097, 1283, 1303, 1307, 1429, 1433, 1453, 1487, 1489, 1493, 1613, 1669, 1699, 1789, 1873, 1877, 1879, 1999

Offset: 1

Reinhard Zumkeller, Sep 07 2004

nonn

Union of A098412 and A098413;

a(n)=A007529(n)+6; either a(n)=A098414(n)+2 or a(n)=A098414(n)+4.

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

Cf. A098424, A098416, A098417.

Transpose[Select[Partition[Prime[Range[350]],3,1],#[[3]]- #[[1]] == 6&]][[3]] (* Harvey P. Dale, Mar 17 2015 *)

is(n)=isprime(n) && isprime(n-6) && (isprime(n-2) || isprime(n-4)) \\ Charles R Greathouse IV, Feb 23 2017

A098413 Greatest members p of prime triples (p-6, p-2, p).

Original entry on oeis.org

13, 19, 43, 73, 103, 109, 199, 229, 283, 313, 463, 619, 829, 859, 883, 1093, 1303, 1429, 1453, 1489, 1669, 1699, 1789, 1873, 1879, 1999, 2089, 2143, 2383, 2689, 2713, 2803, 3169, 3259, 3463, 3469, 3853, 4159, 4519, 4789, 5233, 5419, 5443, 5653, 5659, 5743

Offset: 1

Reinhard Zumkeller, Sep 07 2004

nonn

Subsequence of A046117; a(n) = A073649(n) + 2 = A022005(n) + 6.

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

Cf. A098412, A098415.

[p: p in PrimesUpTo(6500)|IsPrime(p) and IsPrime(p-6) and IsPrime(p-2)]; // Vincenzo Librandi, Dec 26 2010

Transpose[Select[Partition[Prime[Range[800]],3,1],Differences[#] == {4,2}&]][[3]] (* Harvey P. Dale, Aug 21 2013 *)

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

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 *)

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

A214450 Smallest prime p such that n primes exist between the prime triple (p, p+2, p+6) and the next prime triple.

Original entry on oeis.org

5, 857, 311, 17, 31391, 3461, 1427, 12917, 1997, 4517, 41, 20747, 107, 1871, 1487, 4637, 2081, 347, 7877, 23057, 80777, 1091, 18041, 641, 461, 5231, 21017, 881, 4967, 45821, 1607, 15731, 165311, 17027, 35591, 26261, 11777, 8537, 64151, 101111, 82757, 23741

Offset: 0

Michel Lagneau, Jul 18 2012

nonn

			a(3)= 17 because there exists 3 primes 29, 31 and 37 are between (17, 19,23) and (41,43,47).

Cf. A022004, A098412, A089637.

A214450 := proc(n)
local j, hi, lo ;
if n = 0 then
3;
else
for j from 1 do
hi := numtheory[pi]( A022004 (j+1)) ;
lo := numtheory[pi]( A098412 (j)) ;
if hi-lo = n+1 then
return A022004 (j);
end if;
end do:
end if;
end proc: # [Program from R. J. Mathar, adapted for this sequence (see A089637)].

A290706 Greatest of 4 consecutive primes with consecutive gaps 2, 4, 6.

Original entry on oeis.org

29, 53, 239, 359, 653, 1103, 1289, 1439, 1499, 1619, 2699, 3539, 3929, 4013, 4139, 4649, 4799, 4943, 8243, 9473, 10343, 11789, 12119, 13913, 14639, 20759, 21569, 23753, 25589, 26693, 26723, 27749, 27953, 28289, 29033, 31259

Offset: 1

Muniru A Asiru, Aug 09 2017

nonn

All terms = {23, 29} mod 30.

			29 is a member of the sequence because 29 is the greatest of the 4 consecutive primes 17, 19, 23, 29 with consecutive gaps 2, 4, 6.

Cf. A006512, A078847, A098412.

K:=3*10^7+1;; # to get all terms <= K.
P:=Filtered([1,3..K],IsPrime);;    I:=[2,4,6];;
P1:=List([1..Length(P)-1],i->P[i+1]-P[i]);;
P2:=List([1..Length(P)-Length(I)],i->[P1[i],P1[i+1],P1[i+2]]);;
P3:=List(Positions(P2,I),i->P[i+Length(I)]);

for i from 1 to 10^5 do if ithprime(i+1)=ithprime(i)+2 and ithprime(i+2)=ithprime(i)+6 and ithprime(i+3)=ithprime(i)+12 then print(ithprime(i+3)); fi; od;

Select[Prime@ Range@ 3500, NextPrime[#, {1, 2, 3}] == # + {2, 6, 12} &] + 12 (* Giovanni Resta, Aug 09 2017 *)

a(n) = A078847(n) + 12.

cp's OEIS Frontend

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

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A073648 Middle members of prime triples {p, p+2, p+6}.

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A098415 Greatest members r of prime triples (p,q,r) with p

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A098413 Greatest members p of prime triples (p-6, p-2, p).

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

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

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