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.

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A078850 Initial term in sequence of four consecutive primes separated by 3 consecutive differences each <=6 (i.e., when d=2,4 or 6) and forming d-pattern=[4,2,6]; short d-string notation of pattern = [426].

Original entry on oeis.org

67, 1447, 2377, 2707, 5437, 5737, 7207, 9337, 11827, 12037, 19207, 21487, 21517, 23197, 26107, 26947, 28657, 31147, 31177, 35797, 37357, 37567, 42697, 50587, 52177, 65167, 67927, 69997, 71707, 74197, 79147, 81547, 103087, 103387, 106657
Offset: 1

Views

Author

Labos Elemer, Dec 11 2002

Keywords

Comments

Subsequence of A022005. - R. J. Mathar, May 06 2017

Examples

			p=67,67+4=71,67+4+2=73,67+4+2+6=79 are consecutive primes.

Links

Crossrefs

Cf. analogous prime quadruple sequences with various possible {2, 4, 6}-difference-patterns in brackets: A007530[242], A078847[246], A078848[264], A078849[266], A052378[424], A078850[426], A078851[462], A078852[466], A078853[624], A078854[626], A078855[642], A078856[646], A078857[662], A078858[664], A033451[666].

Programs

  • Mathematica
    d = {4, 2, 6}; First /@ Select[Partition[Prime@ Range@ 12000, Length@ d + 1, 1], Differences@ # == d &] (* Michael De Vlieger, May 02 2016 *)

Formula

Primes p = p(i) such that p(i+1)=p+4, p(i+2)=p+4+2, p(i+3)=p+4+2+6.

Extensions

Listed terms verified by Ray Chandler, Apr 20 2009

A382810 Primes p such that p + 6, p + 10 and p + 16 are also primes.

Original entry on oeis.org

7, 13, 31, 37, 73, 97, 157, 223, 373, 433, 1087, 1291, 1423, 1483, 1543, 1861, 1987, 2341, 2383, 2677, 2683, 3313, 3607, 4441, 4507, 4783, 4993, 5641, 5851, 6037, 6961, 7237, 7867, 8731, 9613, 9733, 10723, 13093, 13681, 14143, 14731, 16057, 16411, 16921, 17377
Offset: 1

Views

Author

Alexander Yutkin, Apr 05 2025

Keywords

Comments

The four primes need not be consecutive; otherwise we have the sequence A078856.

Examples

			p=37: 37+6=43, 37+10=47, 37+16=53 -> prime quartet: (37, 43, 47, 53).

Links

Crossrefs

Cf. A000040, A001223, A023200, A031924.
Cf. A078852 [4, 6, 6], A078856 [6, 4, 6], A078858 [6, 6, 4], A033451 [6, 6, 6].

Programs

  • Maple
    q:= p-> andmap(i->isprime(p+i), [0, 6, 10, 16]):
select(q, [$2..20000])[];  # Alois P. Heinz, Apr 05 2025
  • Mathematica
    Select[Prime[Range[2000]],AllTrue[#+{6,10,16},PrimeQ]&] (* James C. McMahon, Apr 13 2025 *)

A078963 Primes p such that the differences between the 5 consecutive primes starting with p are (6,4,6,2).

Original entry on oeis.org

3313, 4993, 5851, 9613, 17971, 23011, 32353, 36913, 45121, 51421, 53881, 54403, 59611, 76243, 90001, 91951, 127591, 130633, 131431, 134353, 140401, 142963, 174061, 229753, 246913, 267661, 303361, 311551, 321313, 340111, 386143, 435553, 465061, 514513, 532993, 618571
Offset: 1

Views

Author

Labos Elemer, Dec 19 2002

Keywords

Comments

Equivalently, primes p such that p, p+6, p+10, p+16 and p+18 are consecutive primes.

Examples

			23011 is in the sequence since 23011, 23017 = 23011 + 6, 23021 = 23011 + 10, 23027 = 23011 + 16 and 23029 = 23011 + 18 are consecutive primes.

Links

Crossrefs

Subsequence of A078856. - R. J. Mathar, May 06 2017
Cf. A001223, A078866, A078867, A078946-A078971, A022006, A022007.

Programs

  • Maple
    L:= [2,3,5,7,11]:
count:= 0: Res:= NULL:
while count < 50 do
  L:= [op(L[2..5]),nextprime(L[5])];
  if L - [L[1]$5] = [0,6,10,16,18] then
    count:= count+1;
    Res:= Res, L[1];
  fi
od:
Res; # Robert Israel, Jun 04 2018
  • Mathematica
    Transpose[Select[Partition[Prime[Range[50000]],5,1],Differences[#]=={6,4,6,2}&]][[1]]  (* Harvey P. Dale, Mar 04 2011 *)
  • PARI
    list(lim) = {my(p1 = 2, p2 = 3, p3 = 5, p4 = 7); forprime(p5 = 11, lim, if(p2 - p1 == 6 && p3 - p2 == 4 && p4 - p3 == 6 && p5 - p4 == 2, print1(p1, ", ")); p1 = p2; p2 = p3; p3 = p4; p4 = p5);} \\ Amiram Eldar, Feb 22 2025

Formula

From Amiram Eldar, Feb 22 2025: (Start)
a(n) == 1 (mod 6).
a(n) == 1 or 13 (mod 30). (End)

Extensions

Edited by Dean Hickerson, Dec 20 2002

A078964 Primes p such that the differences between the 5 consecutive primes starting with p are (6,4,6,6).

Original entry on oeis.org

157, 4441, 6961, 8731, 14731, 16411, 16921, 20107, 25447, 39097, 47287, 47491, 60601, 78157, 78781, 84121, 92347, 104701, 114067, 115321, 128467, 142537, 183571, 186097, 194707, 196171, 253417, 279121, 286477, 297607, 307267, 327001, 350437, 351031, 354307, 357661
Offset: 1

Views

Author

Labos Elemer, Dec 19 2002

Keywords

Comments

Equivalently, primes p such that p, p+6, p+10, p+16 and p+22 are consecutive primes.

Examples

			157 is in the sequence since 157, 163 = 157 + 6, 167 = 157 + 10, 173 = 157 + 16 and 179 = 157 + 22 are consecutive primes.

Links

Crossrefs

Subsequence of A078856. - R. J. Mathar, May 06 2017
Cf. A001223, A078866, A078867, A078946-A078971, A022006, A022007.

Programs

  • Mathematica
    Select[Partition[Prime[Range[50000]], 5, 1], Differences[#] == {6,4,6,6} &][[;;, 1]] (* Amiram Eldar, Feb 22 2025 *)
  • PARI
    list(lim) = {my(p1 = 2, p2 = 3, p3 = 5, p4 = 7); forprime(p5 = 11, lim, if(p2 - p1 == 6 && p3 - p2 == 4 && p4 - p3 == 6 && p5 - p4 == 6, print1(p1, ", ")); p1 = p2; p2 = p3; p3 = p4; p4 = p5);} \\ Amiram Eldar, Feb 22 2025

Formula

From Amiram Eldar, Feb 22 2025: (Start)
a(n) == 1 (mod 6).
a(n) == 1 or 7 (mod 30). (End)

Extensions

Edited by Dean Hickerson, Dec 20 2002

A079018 Suppose p and q = p+16 are primes. Define the difference pattern of (p,q) to be the successive differences of the primes in the range p to q. There are 17 possible difference patterns, namely [16], [4,12], [6,10], [10,6], [12,4], [4,2,10], [4,6,6], [4,8,4], [6,4,6], [6,6,4], [10,2,4], [4,2,4,6], [4,2,6,4], [4,6,2,4], [6,4,2,4], [4,2,4,2,4], [2,2,4,2,4,2]. Sequence gives smallest value of p for each difference pattern, sorted by magnitude.

Original entry on oeis.org

3, 7, 13, 31, 43, 67, 73, 151, 181, 211, 241, 277, 331, 463, 487, 1597, 1831
Offset: 1

Views

Author

Labos Elemer, Jan 24 2003

Keywords

Examples

			p=181, q=197 has difference pattern [10,2,4] and {181,191,193,197} is the corresponding consecutive prime 4-tuple.

Crossrefs

A022008(1)=7, A078952(1)=13, A078852(1)=73, A078953(1)=67, A078954(1)=1597, A078961(1)=31, A078856(1)=73, A078858(1)=151, A031934(1)=A000230(8)=1831.
Cf. A079016-A079024.
Previous Showing 11-15 of 15 results.

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