A078561 -id:A078561 - cp's OEIS frontend

A286313 Union of A078561 and A078562.

19, 31, 43, 61, 73, 79, 127, 157, 163, 229, 271, 349, 373, 379, 433, 439, 499, 607, 643, 673, 733, 751, 937, 967, 1009, 1093, 1213, 1279, 1291, 1429, 1489, 1543, 1549, 1597, 1609, 1657, 1777, 1861, 1987, 2131, 2203, 2287, 2341, 2347, 2371, 2383, 2389

Offset: 1

Zak Seidov, May 06 2017

nonn

Number of terms among first 10^k primes, k=1..8:
0, 1, 17, 105, 646, 4385, 31721, 240346, 1884832.
E.g., k=1, first 10 primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, and only 19 is a term of the sequence. - Zak Seidov, May 08 2017
Primes p such that prime(p+2) = p + 10. - Harvey P. Dale, Jan 13 2022

Harvey P. Dale, Table of n, a(n) for n = 1..2000
R. J. Mathar, Table of Prime Gap Constellations

Cf. A078561 and A078562.

select(p -> isprime(p) and isprime(p+10) and (isprime(p+4) xor isprime(p+6)), [seq(i,i=5..10000,2)]); # Robert Israel, May 08 2017

Select[Prime[Range[1000]], NextPrime[#, 2] == # + 10 &]
Select[Partition[Prime[Range[400]],3,1],#[[1]]+10==#[[3]]&][[All,1]] (* Harvey P. Dale, Jan 13 2022 *)

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

A078562 p, p+6 and p+10 are consecutive primes.

Original entry on oeis.org

31, 61, 73, 157, 271, 373, 433, 607, 733, 751, 1291, 1543, 1657, 1777, 1861, 1987, 2131, 2287, 2341, 2371, 2383, 2467, 2677, 2791, 2851, 3181, 3313, 3607, 3691, 4441, 4507, 4723, 4993, 5407, 5431, 5521, 5563, 5641, 5683, 5851, 6037, 6211, 6571, 6961

Offset: 1

Labos Elemer, Dec 10 2002

nonn

Subsequence of A031924. - R. J. Mathar, Jun 15 2013

			Between p and p+10 the difference-pattern is [64] like e.g. for p=31: 31(6)37(4)41.

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

Cf. analogous inter-prime d-patterns with d<=6: A022004[24], A022005[42], A049437[26], A049438[62], A078561[46], A078562[64], A047948[66].

Transpose[Select[Partition[Prime[Range[1000]],3,1],#[[3]]-#[[1]]==10&&#[[2]]-#[[1]]==6&]][[1]] (* Harvey P. Dale, Dec 09 2010 *)

cp's OEIS Frontend

A286313 Union of A078561 and A078562.

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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].

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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].

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A078562 p, p+6 and p+10 are consecutive primes.

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