A078852 -id:A078852 - cp's OEIS frontend

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].

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

Labos Elemer, Dec 11 2002

nonn

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

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

R. J. Mathar, Table of n, a(n) for n = 1..1000

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].

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

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

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

Alexander Yutkin, Apr 05 2025

nonn

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

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

Robert Israel, Table of n, a(n) for n = 1..2500

Cf. A000040, A001223, A023200, A031924.

Cf. A078852 [4, 6, 6], A078856 [6, 4, 6], A078858 [6, 6, 4], A033451 [6, 6, 6].

q:= p-> andmap(i->isprime(p+i), [0, 6, 10, 16]):
select(q, [$2..20000])[];  # Alois P. Heinz, Apr 05 2025

Select[Prime[Range[2000]],AllTrue[#+{6,10,16},PrimeQ]&] (* James C. McMahon, Apr 13 2025 *)

A078869 Number of n-tuples with elements in {2,4,6} which can occur as the differences between n+1 consecutive primes > n+1. (Values of a(11), ..., a(18) are conjectured to be correct, but are only known to be upper bounds.)

Original entry on oeis.org

3, 7, 15, 26, 38, 48, 67, 92, 105, 108, 109, 118, 130, 128, 112, 80, 36, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Labos Elemer, Dec 19 2002

nonn

The ">n+1" rules out n-tuples like (2,2), which only occurs for the primes 3, 5, 7. All terms from a(19) on equal 0.

An n-tuple (a_1,a_2,...,a_n) is counted iff the partial sums 0, a_1, a_1+a_2, ..., a_1+...+a_n do not contain a complete residue system (mod p) for any prime p.

Eric Weisstein's World of Mathematics, k-Tuple Conjecture

The 26 4-tuples and 38 5-tuples are in A078868 and A078870. Cf. A001359, A008407, A029710, A031924, A022004-A022007, A078852, A078858, A078946-A078969, A020497.

test[tuple_] := Module[{r, sums, i, j}, r=Length[tuple]; sums=Prepend[tuple.Table[If[j>=i, 1, 0], {i, 1, r}, {j, 1, r}], 0]; For[i=1, Prime[i]<=r+1, i++, If[Length[Union[Mod[sums, Prime[i]]]]==Prime[i], Return[False]]]; True]; tuples[0]={{}}; tuples[n_] := tuples[n]=Select[Flatten[Outer[Append, tuples[n-1], {2, 4, 6}, 1], 1], test]; a[n_] := Length[tuples[n]]

Edited by Dean Hickerson, Dec 20 2002

A078956 Primes p such that the differences between the 5 consecutive primes starting with p are (4,6,6,2).

Original entry on oeis.org

43, 163, 643, 1213, 2953, 4003, 7573, 11923, 14533, 25453, 26683, 26713, 29863, 41593, 48523, 61543, 68473, 150193, 151153, 172423, 206803, 227593, 290023, 302563, 338563, 343813, 346543, 428023, 527053, 529033, 540373, 547483, 551713, 570403, 577513, 622603, 628993

Offset: 1

Labos Elemer, Dec 19 2002

nonn

Equivalently, primes p such that p, p+4, p+10, p+16 and p+18 are consecutive primes.

All terms == 13 (mod 30). - Robert Israel, Oct 17 2023

			43 is in the sequence since 43, 47 = 43 + 4, 53 = 43 + 10, 59 = 43 + 16 and 61 = 43 + 18 are consecutive primes.

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000 (first 1000 terms from Robert Israel)

Subsequence of A078852. - R. J. Mathar, May 06 2017

Cf. A001223, A078866, A078867, A078946-A078969, A022006, A022007.

L:= [0$5]:
p:= 1: R:= NULL: count:= 0:
while count < 100 do
  p:= nextprime(p);
  L:= [L[2],L[3],L[4],L[5],p];
  if L -~ L[1] = [0, 4, 10, 16, 18] then
    count:= count+1;
    R:= R, L[1];
  fi
od:
R; # Robert Israel, Oct 17 2023

Select[Partition[Prime[Range[50000]],5,1],Differences[#]=={4,6,6,2}&][[All,1]] (* Harvey P. Dale, Jan 23 2021 *)

list(lim) = {my(p1 = 2, p2 = 3, p3 = 5, p4 = 7); forprime(p5 = 11, lim, if(p2 - p1 == 4 && p3 - p2 == 6 && p4 - p3 == 6 && p5 - p4 == 2, print1(p1, ", ")); p1 = p2; p2 = p3; p3 = p4; p4 = p5);} \\ Amiram Eldar, Feb 21 2025

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

Labos Elemer, Jan 24 2003

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

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.

cp's OEIS Frontend

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].

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

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A078869 Number of n-tuples with elements in {2,4,6} which can occur as the differences between n+1 consecutive primes > n+1. (Values of a(11), ..., a(18) are conjectured to be correct, but are only known to be upper bounds.)

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A078956 Primes p such that the differences between the 5 consecutive primes starting with p are (4,6,6,2).

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