A276826 -id:A276826 - cp's OEIS frontend

A022009 Initial members of prime septuplets (p, p+2, p+6, p+8, p+12, p+18, p+20).

11, 165701, 1068701, 11900501, 15760091, 18504371, 21036131, 25658441, 39431921, 45002591, 67816361, 86818211, 93625991, 124716071, 136261241, 140117051, 154635191, 162189101, 182403491, 186484211, 187029371, 190514321, 198453371

Offset: 1

Warut Roonguthai

nonn

All terms are congruent to 11 (modulo 210). - Matt C. Anderson, May 26 2015
Also the terms k of A276848 for which k == 1 (mod 10), see the comment in A276848 and A276826. All terms are obviously also congruent to 11 (modulo 30). - Vladimir Shevelev, Sep 21 2016
See A343637 for the least prime septuplets > 10^n, n >= 0. - M. F. Hasler, Aug 04 2021

Dana Jacobsen, Table of n, a(n) for n = 1..10000 (first 1000 terms from Matt C. Anderson)
Matt C. Anderson, table of prime k-tuplets.
Tony Forbes and Norman Luhn, Patterns of prime k-tuplets & the Hardy-Littlewood constants.
Norman Luhn, 1 million terms, zipped archive.
Vladimir Shevelev and Peter J. C. Moses, Constellations of primes generated by twin primes, arXiv:1610.03385 [math.NT], 2016.
Eric Weisstein's World of Mathematics, Prime Constellation.

Cf. A022010 (septuplets of the second type), A182387, A276826, A276848, A343637 (septuplet following 10^n).

[p: p in PrimesUpTo(2*10^8) | forall{p+r: r in [2,6,8,12,18,20] | IsPrime(p+r)}]; // Vincenzo Librandi, Oct 01 2015

Transpose[Select[Partition[Prime[Range[10400000]],7,1],Differences[#] == {2,4,2,4,6,2}&]][[1]] (* Harvey P. Dale, Jul 13 2014 *)
Select[Prime[Range[2 10^8]], Union[PrimeQ[# + {2, 6, 8, 12, 18, 20}]] == {True} &] (* Vincenzo Librandi, Oct 01 2015 *)

nextcomposite(n)=if(n<4, return(4)); n=ceil(n); if(isprime(n), n+1, n)
   is(n)=if(n%30!=11 || !isprime(n) || !isprime(n+2), return(0)); my(p=n, q=n+2, k=2, f); while(p!=q && q-p<7, f=if(isprime(k++), nextprime, nextcomposite); p=f(p+1); q=f(q+1)); p==q \\ Charles R Greathouse IV, Sep 30 2016

select( {is_A022009(n)=n%210==11&&!foreach([20,18,12,8,6,2,0],d,isprime(n+d)||return)}, [11+k*210|k<-[0..10^5]]) \\ M. F. Hasler, Aug 04 2021

use ntheory ":all"; say for sieve_prime_cluster(1,1e9, 2,6,8,12,18,20); # Dana Jacobsen, Sep 30 2015

a(n) = 210*A182387(n) + 11. - Hugo Pfoertner, Nov 18 2022

A277118 For a lesser p of twin primes, let B_k be A159559, but with initial term k; then a(n) is the smallest m such that B_(p+2)(m)-B_p(m)>6, where p = A001359(n-1), or a(n) = 0 if there is no such m.

Original entry on oeis.org

0, 13, 0, 0, 0, 9, 0, 11, 11, 5, 3, 15, 3, 7, 3, 0, 3, 0, 3, 5, 7, 3, 11, 5, 3, 5, 11, 3, 9, 3, 3, 7, 3, 5, 5, 3, 5, 3, 5, 11, 3, 5, 0, 0, 5, 5, 7, 5, 13, 7, 0, 5, 3, 3, 3, 3, 7, 3, 3, 3, 5, 3, 7, 3, 3, 0, 3, 5, 5, 3, 11, 11, 5, 3, 5, 7, 5, 3, 0, 3, 3, 3, 3, 3

Offset: 2

Vladimir Shevelev and Peter J. C. Moses, Sep 30 2016

nonn

Theorem: a(n) takes only the values 0, 3, 5, 7, 9, 11, 13, 15, and 17.

Peter J. C. Moses, Table of n, a(n) for n = 2..5001
Vladimir Shevelev, "Nearest" twin primes, Post to seqfan, Sep 21 2016.
Vladimir Shevelev, Peter J. C. Moses, Constellations of primes generated by twin primes, arXiv:1610.03385 [math.NT], 2016.

Cf. A022009, A159559, A229019, A276676, A276703, A276767, A276826, A276831, A276848.

nextcomposite(n)=if(n<4, return(4)); n=ceil(n); if(isprime(n), n+1, n)
do(p)=my(a=p,b=p+2,f); for(n=3,17, f=if(isprime(n), nextprime, nextcomposite); a=f(a+1); b=f(b+1); if(b-a > 6, return(n))); 0
p=2; forprime(q=3,1e3, if(q-p==2, print1(do(p)", ")); p=q) \\ Charles R Greathouse IV, Oct 17 2016

a(n) = 3 on a subsequence of measure 1. - Charles R Greathouse IV, Oct 17 2016

A276848 For a lesser p of twin primes, let B_(p+2) and B_p be sequences defined as A159559, but with initial terms p+2 and p respectively. The sequence lists p for which all differences B_(p+2)(n)-B_p(n)<=6.

Original entry on oeis.org

3, 11, 17, 29, 59, 227, 269, 1277, 1289, 1607, 2129, 2789, 3527, 3917, 4637, 4787, 5639, 8999, 13679, 14549, 18119, 27737, 36779, 38447, 39227, 44267, 62129, 71327, 75989, 80669, 83219, 88799, 93479, 97367, 99707, 113147, 113159, 115769, 122027, 122387, 124337, 124769, 132749, 150209, 160079

Offset: 1

Vladimir Shevelev, Sep 21 2016

nonn

B_(p+2)(n) - B_p(n) < 6 for all n >= 2 if and only if p = 3.

It is astonishing that, although terms a(n) == 7 or 9 (mod 10) occur often, the first terms a(n)==1 (mod 10) are 11, 165701, ... (cf. A022009). This phenomenon is explained in the Shevelev link.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Vladimir Shevelev, "nearest" twin primes, Post to seqfan, Sep 21 2016.
Vladimir Shevelev, Peter J. C. Moses, Constellations of primes generated by twin primes, arXiv:1610.03385 [math.NT], 2016.

Cf. A001359, A159559, A229019, A276676, A276703, A276767, A276826, A276831, A022009.

nextcomposite(n)=if(n<4, return(4)); n=ceil(n); if(isprime(n), n+1, n)
is(n)=if(!isprime(n) || !isprime(n+2), return(0)); my(p=n,q=n+2,k=2,f); while(p!=q && q-p<7, f=if(isprime(k++),nextprime,nextcomposite); p=f(p+1); q=f(q+1)); p==q \\ Charles R Greathouse IV, Sep 21 2016

More terms from Peter J. C. Moses, Sep 21 2016

A276831 For a lesser p=A001359(n-1), n>=2, of twin primes, let B_k be the sequence defined as A159559 but with initial term k; a(n) is the smallest m such that B_(p+2)(m)-B_p(m) = max_{t>=2} (B_(p+2)(t)-B_p(t)).

Original entry on oeis.org

5, 17, 11, 5, 3, 17, 3, 11, 11, 5, 31, 107, 13, 333, 17, 5, 3, 3, 281, 5, 997, 3, 487, 659, 5178, 5, 15, 3, 23, 53, 13, 1567, 13, 13, 181, 3, 5, 443, 37, 21, 19, 11, 5, 3, 5, 5, 7, 20786, 13, 7, 5, 21, 3, 5, 17, 61, 31, 23, 7, 3, 11, 5, 11, 5, 3, 3, 157, 37

Offset: 2

Vladimir Shevelev, Sep 20 2016

nonn

			Let n=2, p=A001359(1)=3. Then B_3(2)=3, B_3(3)=5, B_3(4)=6, B_3(5)=7, B_3(6)=8, B_3(7)=11, B_3(8)=12, B_3(9)=14, B_3(10)=15, B_3(11)=17;
Further, B_5(2)=5, B_5(3)=7, B_5(4)=8, B_5(5)=11, B_5(6)=12, B_5(7)=13, B_5(8)=14, B_5(9)=15, B_5(10)=16, B_5(11)=17 and, beginning with t=11,
B_3 merges with B_5. So, max(B_5(t)-B_3(t))=4 reaching at t=5 and t=6.
Thus a(2)=min(5,6)=5.

Vladimir Shevelev, Peter J. C. Moses, Constellations of primes generated by twin primes, arXiv:1610.03385 [math.NT], 2016.

Cf. A001359, A159559, A229019, A276676, A276703, A276767, A276826.

B_(p+2)(a(n)) - B_p(a(n)) = A276826(n).

More terms from Peter J. C. Moses, Sep 20 2016

cp's OEIS Frontend

A022009 Initial members of prime septuplets (p, p+2, p+6, p+8, p+12, p+18, p+20).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Perl

Formula

A277118 For a lesser p of twin primes, let B_k be A159559, but with initial term k; then a(n) is the smallest m such that B_(p+2)(m)-B_p(m)>6, where p = A001359(n-1), or a(n) = 0 if there is no such m.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A276848 For a lesser p of twin primes, let B_(p+2) and B_p be sequences defined as A159559, but with initial terms p+2 and p respectively. The sequence lists p for which all differences B_(p+2)(n)-B_p(n)<=6.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Extensions

A276831 For a lesser p=A001359(n-1), n>=2, of twin primes, let B_k be the sequence defined as A159559 but with initial term k; a(n) is the smallest m such that B_(p+2)(m)-B_p(m) = max_{t>=2} (B_(p+2)(t)-B_p(t)).

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

Extensions