A052378 -id:A052378 - cp's OEIS frontend

A022007 Initial members of prime 5-tuples (p, p+4, p+6, p+10, p+12).

7, 97, 1867, 3457, 5647, 15727, 16057, 19417, 43777, 79687, 88807, 101107, 257857, 266677, 276037, 284737, 340927, 354247, 375247, 402757, 419047, 427237, 463447, 470077, 626617, 666427, 736357, 823717, 855727, 959467, 978067, 1022377, 1043587, 1068247

Offset: 1

Warut Roonguthai

Subsequence of A052378. - R. J. Mathar, Feb 11 2013
All terms are congruent to 7 (modulo 30). - Matt C. Anderson, May 22 2015
This sequence is related to the admissible prime 5-tuple (0, 4, 6, 10, 12) because the sequence [1, 2, 3, 1, 2, repeat(1)] gives for n >= 1 the smallest element of RS0(A000040(n)) (the smallest nonnegative complete residue systems modulo prime(n)) which defines a residue class containing none of the 5-tuple members. This 5-tuple is one of two prime constellations of diameter 12. The other one is (0, 2, 6, 8, 12) with initial members given in A022006. See the Wikipedia and Weisstein pages. - Wolfdieter Lang, Oct 06 2017

			Admissibility guaranteeing sequence [1, 2, 3, 1, 2, repeat(1)] examples: the only residue class modulo prime(3) = 5 which contains none of the 5-tuple (0, 4, 6, 10, 12) members is 3 (mod 5). For prime(5) = 11 the first class is 2 (mod 11); the others are 3, 5, 7, 8, 9 (mod 11). - _Wolfdieter Lang_, Oct 06 2017

Zak Seidov, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
T. Forbes and Norman Luhn, Prime k-tuplets.
Norman Luhn, Table of n, a(n) for n = 1..1000000.
Eric Weisstein's World of Mathematics, Prime Constellation.
Wikipedia, Prime k-tuple.

Cf. A022006, A089157.

[p: p in PrimesUpTo(2*10^6) | IsPrime(p+4) and IsPrime(p+6) and IsPrime(p+10)and IsPrime(p+12)]; // Vincenzo Librandi, Aug 23 2015

Transpose[Select[Partition[Prime[Range[76000]], 5, 1], Differences[#] == {4, 2, 4, 2} &]][[1]] (* Harvey P. Dale, Aug 16 2014 *)

forprime(p=2,1e5,if(isprime(p+4)&&isprime(p+6)&&isprime(p+10)&&isprime(p+12),print1(p", "))) \\ Charles R Greathouse IV, Jul 15 2011

use ntheory ":all"; say for sieve_prime_cluster(1,1e7, 4,6,10,12); # Dana Jacobsen, Sep 30 2015

a(n) = 7 + 30*A089157(n). - Zak Seidov, Nov 01 2011

A078847 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 pattern = [2, 4, 6]; short notation = [246] d-pattern.

Original entry on oeis.org

17, 41, 227, 347, 641, 1091, 1277, 1427, 1487, 1607, 2687, 3527, 3917, 4001, 4127, 4637, 4787, 4931, 8231, 9461, 10331, 11777, 12107, 13901, 14627, 20747, 21557, 23741, 25577, 26681, 26711, 27737, 27941, 28277, 29021, 31247, 32057, 32297

Offset: 1

Labos Elemer, Dec 11 2002

nonn

Subsequence of A022004. - R. J. Mathar, Feb 10 2013

a(n) + 12 is the greatest term in the sequence of 4 consecutive primes with 3 consecutive gaps 2, 4, 6. - Muniru A Asiru, Aug 03 2017

			17, 17+2 = 19, 17+2+4 = 23, 17+2+4+6 = 29 are consecutive primes.

Zak Seidov, Table of n, a(n) for n = 1..2000

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].

Cf. A190814[2,4,6,8], A190817[2,4,6,8,10], A190819[2,4,6,8,10,12], A190838[2,4,6,8,10,12,14]

d = Differences[Prime[Range[10000]]]; Prime[Flatten[Position[Partition[d, 3, 1], {2, 4, 6}]]] (* T. D. Noe, May 23 2011 *)
Transpose[Select[Partition[Prime[Range[10000]],4,1],Differences[#] == {2,4,6}&]][[1]] (* Harvey P. Dale, Aug 07 2013 *)

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

Listed terms verified by Ray Chandler, Apr 20 2009

Additional cross references from Harvey P. Dale, May 10 2014

A078857 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=[6, 6,2]; short d-string notation of pattern = [662].

Original entry on oeis.org

47, 167, 257, 557, 587, 647, 1217, 2957, 4007, 6257, 6857, 7577, 10847, 11927, 14537, 16217, 17477, 19457, 24407, 25457, 26687, 26717, 29867, 41507, 41597, 48527, 51407, 54617, 56087, 60077, 61547, 68477, 75527, 82457, 84047, 94427, 101267

Offset: 1

Labos Elemer, Dec 11 2002

nonn

Subsequence of A047948. - R. J. Mathar, Feb 11 2013

			p=47,47+6=53,47+6+6=59,47+6+6+2=61 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].

Select[Partition[Prime[Range[10000]],4,1],Differences[#]=={6,6,2}&][[All,1]] (* Harvey P. Dale, Apr 29 2017 *)

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

Listed terms verified by Ray Chandler, Apr 20 2009

A078858 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 = [6, 6, 4]; short d-string notation of pattern = [664].

Original entry on oeis.org

151, 367, 601, 727, 2281, 2671, 3307, 4987, 5557, 10651, 12967, 13171, 15907, 18217, 18427, 20101, 20341, 24091, 27061, 28591, 30097, 30307, 31321, 32491, 35311, 37951, 41941, 42181, 42391, 45751, 52951, 53617, 55201, 56767, 59107, 65407

Offset: 1

Labos Elemer, Dec 11 2002

nonn

Subsequence of A047948. - R. J. Mathar, Feb 11 2013

			p=151, 151+6 = 157, 151+6+6 = 163, 151+6+6+4 = 167 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].

Transpose[Select[Partition[Prime[Range[6600]],4,1],Differences[#] == {6,6,4}&]][[1]] (* Harvey P. Dale, Nov 04 2011 *)

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

Listed terms verified by Ray Chandler, Apr 20 2009

A078854 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=[6, 2,6]; short d-string notation of pattern = [626].

Original entry on oeis.org

23, 53, 263, 563, 593, 1223, 1283, 1613, 2333, 2543, 3533, 4013, 4643, 5843, 6263, 6353, 6563, 10853, 11483, 14543, 15263, 17483, 19073, 19373, 19463, 23663, 26723, 29123, 32363, 34253, 41603, 48473, 49193, 49523, 51413, 51473, 71333, 75983

Offset: 1

Labos Elemer, Dec 11 2002

nonn

Subsequence of A049438. - R. J. Mathar, May 06 2017

			p=23,23+6=29,23+6+2=31,23+6+2+6=37 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].

Transpose[Select[Partition[Prime[Range[7500]],4,1],Differences[#]=={6,2,6}&]][[1]] (* Harvey P. Dale, Apr 17 2015 *)

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

Listed terms verified by Ray Chandler, Apr 20 2009

A078855 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=[6, 4,2]; short d-string notation of pattern = [642].

Original entry on oeis.org

31, 61, 271, 607, 1291, 1657, 1777, 1861, 1987, 2131, 2371, 2677, 2791, 4507, 5407, 5431, 5641, 7867, 9001, 11821, 13681, 14551, 17377, 18121, 18301, 20347, 21481, 22147, 24097, 27271, 32707, 35521, 36781, 37561, 41221, 41947, 42397, 42451

Offset: 1

Labos Elemer, Dec 11 2002

nonn

Subsequence of A078562. - R. J. Mathar, May 06 2017

			p=31,31+6=37,31+6+4=41,31+6+4+2=43 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].

Transpose[Select[Partition[Prime[Range[4500]],4,1],Differences[#] == {6,4,2}&]][[1]] (* Harvey P. Dale, Feb 10 2015 *)

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

Listed terms verified by Ray Chandler, Apr 20 2009

A078848 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=[2,6,4]; short d-string notation of pattern = [264].

Original entry on oeis.org

29, 59, 71, 269, 431, 1289, 2129, 2339, 2381, 2789, 4721, 5519, 5639, 5849, 6569, 6959, 8999, 10091, 13679, 14549, 16649, 16691, 18119, 19379, 19751, 21491, 25931, 27689, 27791, 28619, 31181, 32369, 32561, 32831, 36779, 41609, 43961, 45119

Offset: 1

Labos Elemer, Dec 11 2002

nonn

Subsequence of A049437. - R. J. Mathar, Feb 10 2013

			29, 29+2=31, 29+2+6=37, 29+2+6+4=41 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 = {2, 6, 4}; First /@ Select[Partition[Prime@ Range[10^4], Length@ d + 1, 1], Differences@ # == d &] (* Michael De Vlieger, May 02 2016 *)
Select[Partition[Prime[Range[4700]],4,1],Differences[#]=={2,6,4}&][[All,1]] (* Harvey P. Dale, Mar 08 2020 *)

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

Listed terms verified by Ray Chandler, Apr 20 2009

Typo in example corrected by Michel Marcus, Dec 28 2013

A078851 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, 6, 2]; short d-string notation of pattern = [462].

Original entry on oeis.org

19, 127, 229, 1009, 1279, 1597, 1609, 2539, 3319, 3529, 3907, 3919, 4639, 4789, 4999, 5839, 5857, 7477, 7537, 8419, 9619, 12097, 12907, 13327, 15259, 15877, 17569, 17977, 19069, 22027, 23017, 24967, 27739, 28537, 32359, 33577, 36919, 38317

Offset: 1

Labos Elemer, Dec 11 2002

nonn

Subsequence of A078561. - R. J. Mathar, May 06 2017

			p=19,19+4=23,19+4+6=29,19+4+6+2=31 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].

Select[Prime@ Range[10^4], Differences@ Prime@ Range[#, # + 3] &@ PrimePi@ # == {4, 6, 2} &] (* Michael De Vlieger, Jul 02 2016 *)

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

Listed terms verified by Ray Chandler, Apr 20 2009

A078852 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, 6,6]; short d-string notation of pattern = [466].

Original entry on oeis.org

43, 163, 643, 937, 967, 1093, 1213, 2953, 4003, 4447, 6967, 7573, 8737, 9463, 10243, 10597, 11923, 12487, 12637, 13033, 14533, 14737, 15787, 16087, 16417, 16477, 16927, 17317, 17467, 20113, 22063, 25453, 26683, 26713, 27763, 29863, 32983

Offset: 1

Labos Elemer, Dec 11 2002

nonn

Subsequence of A078561. - R. J. Mathar, Feb 11 2013

			p=43,43+4=47,43+4+6=53,43+4+6+6=59 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].

Transpose[Select[Partition[Prime[Range[4000]],4,1],Differences[#]=={4,6,6}&]][[1]] (* Harvey P. Dale, Dec 15 2015 *)

isok(n) = isprime(n) && (nextprime(n+1) == (n+4)) && (nextprime(n+5) == (n+10)) && (nextprime(n+11) == (n+16)) \\ Michel Marcus, Jul 23 2013

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

Listed terms verified by Ray Chandler, Apr 20 2009

A078856 Initial term in sequence of four consecutive primes whose consecutive differences have d-pattern = [6, 4, 6]; short d-string notation for pattern = [646].

Original entry on oeis.org

73, 157, 373, 433, 1543, 2341, 2383, 3313, 3607, 4441, 4993, 5851, 6037, 6961, 7237, 8731, 9613, 9733, 10723, 13093, 14143, 14731, 16411, 16921, 17971, 18787, 20107, 21391, 23011, 23593, 25111, 25237, 25447, 27793, 30103, 30697, 32353, 32563

Offset: 1

Labos Elemer, Dec 11 2002

nonn

			p=73, 73 + 6 = 79, 73 + 6 + 4 = 83, 73 + 6 + 4 + 6 = 89 are consecutive primes.

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

Subsequence of A078562.

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].

N:=10^4: # to get all terms <= N.
Primes:=select(isprime,[seq(i,i=3..N+16,2)]):
Primes[select(t->[Primes[t+1]-Primes[t], Primes[t+2]-Primes[t+1],
Primes[t+3]-Primes[t+2]]=[6,4,6], [$1..nops(Primes)-3])]; # Muniru A Asiru, Aug 04 2017

Transpose[Select[Partition[Prime[Range[10000]],4,1],Differences[#]=={6,4,6}&]][[1]] (* Harvey P. Dale, Apr 22 2014 *)

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

Listed terms verified by Ray Chandler, Apr 20 2009

Name simplified by Michel Marcus, Aug 11 2017

cp's OEIS Frontend

A022007 Initial members of prime 5-tuples (p, p+4, p+6, p+10, p+12).

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A078847 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 pattern = [2, 4, 6]; short notation = [246] d-pattern.

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A078857 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=[6, 6,2]; short d-string notation of pattern = [662].

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A078858 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 = [6, 6, 4]; short d-string notation of pattern = [664].

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A078854 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=[6, 2,6]; short d-string notation of pattern = [626].

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A078855 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=[6, 4,2]; short d-string notation of pattern = [642].

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A078848 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=[2,6,4]; short d-string notation of pattern = [264].

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