cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A270998 Table read by rows: list of prime 5-tuples of the form (p, p+2, p+6, p+8, p+12).

Original entry on oeis.org

5, 7, 11, 13, 17, 11, 13, 17, 19, 23, 101, 103, 107, 109, 113, 1481, 1483, 1487, 1489, 1493, 16061, 16063, 16067, 16069, 16073, 19421, 19423, 19427, 19429, 19433, 21011, 21013, 21017, 21019, 21023, 22271, 22273, 22277, 22279, 22283, 43781, 43783, 43787, 43789, 43793
Offset: 1

Views

Author

Arkadiusz Wesolowski, Jul 12 2016

Keywords

Comments

A prime 5-tuple is a constellation of five successive primes with distance 12, and is of the form (p, p+2, p+6, p+8, p+12) or (p, p+4, p+6, p+10, p+12).
Initial members p (other than 5) of prime 5-tuples of the form (p, p+2, p+6, p+8, p+12) are congruent to 11 or 101 (mod 210).
Also called prime 5-tuples of the first kind.

Links

Crossrefs

Cf. A022006, A270999.

Programs

  • Magma
    lst:=[]; for p in [5..43781 by 2] do if p eq 5 xor p mod 210 in {11, 101} then if IsPrime(p) then t:=[c: c in [p+2..p+12] | IsPrime(c)]; if #t eq 4 then lst:=lst cat [p] cat t; end if; end if; end if; end for; lst;
  • Mathematica
    m = {0, 2, 6, 8, 12}; Union@ Flatten@ Map[# + m &, Select[Prime@ Range[10^4], Times @@ Boole@ PrimeQ[# + m] == 1 &]] (* Michael De Vlieger, Jul 13 2016 *)
Select[Partition[Prime[Range[5000]],5,1],Differences[#]=={2,4,2,4}&]// Flatten (* Harvey P. Dale, Jul 27 2020 *)

Formula

a(5*n-4) = A022006(n).

A271000 Table read by rows: list of prime sextuplets (p, p+4, p+6, p+10, p+12, p+16).

Original entry on oeis.org

7, 11, 13, 17, 19, 23, 97, 101, 103, 107, 109, 113, 16057, 16061, 16063, 16067, 16069, 16073, 19417, 19421, 19423, 19427, 19429, 19433, 43777, 43781, 43783, 43787, 43789, 43793, 1091257, 1091261, 1091263, 1091267, 1091269, 1091273, 1615837, 1615841, 1615843, 1615847, 1615849, 1615853
Offset: 1

Views

Author

Arkadiusz Wesolowski, Jul 12 2016

Keywords

Comments

A prime sextuplet is a constellation of six successive primes with distance 16, and is of the form (p, p+4, p+6, p+10, p+12, p+16).
Initial members p (other than 7) of prime sextuplets are congruent to 97 (mod 210). - Ash, David, Aug 04 2017

Links

Crossrefs

Cf. A022008, A270999.

Programs

  • Magma
    lst:=[]; for p in [5..1615837 by 2] do if p le 7 xor p mod 210 eq 97 then if IsPrime(p) then t:=[c: c in [p+4..p+16] | IsPrime(c)]; if #t eq 5 then lst:=lst cat [p] cat t; end if; end if; end if; end for; lst;
  • Mathematica
    m = {0, 4, 6, 10, 12, 16}; Union@ Flatten@ Map[# + m &, Select[Prime@ Range[2*10^5], Times @@ Boole@ PrimeQ[# + m] == 1 &]] (* Michael De Vlieger, Jul 13 2016 *)

Formula

a(6*n-5) = A022008(n).

A275516 Table read by rows: list of prime triples of the form (p, p+4, p+6).

Original entry on oeis.org

7, 11, 13, 13, 17, 19, 37, 41, 43, 67, 71, 73, 97, 101, 103, 103, 107, 109, 193, 197, 199, 223, 227, 229, 277, 281, 283, 307, 311, 313, 457, 461, 463, 613, 617, 619, 823, 827, 829, 853, 857, 859, 877, 881, 883, 1087, 1091, 1093, 1297, 1301, 1303, 1423, 1427, 1429
Offset: 1

Views

Author

Arkadiusz Wesolowski, Jul 31 2016

Keywords

Comments

A prime triple is a set of three prime numbers of the form (p, p+2, p+6) or (p, p+4, p+6).
Initial members p of prime triples of the form (p, p+4, p+6) are congruent to 7 or 13 (mod 30).
Also called prime triples of the second kind.

Examples

			The table starts:
7, 11, 13;
13, 17, 19;
37, 41, 43;
...

Links

Crossrefs

Cf. A022005, A270999, A271000, A275515.

Programs

  • Magma
    &cat[[p, p+4, p+6]: p in PrimesUpTo(1423) | p mod 30 in {7, 13} and IsPrime(p+4) and IsPrime(p+6)];
  • Mathematica
    Prime@ Range[#, # + 2] &@ PrimePi@ Select[Prime@ Range@ 240, Times @@ Boole@ PrimeQ[# + {4, 6}] > 0 &] // Flatten (* Michael De Vlieger, Aug 02 2016 *)

Formula

a(3*n-2) = A022005(n).
Showing 1-3 of 3 results.

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