A033451 -id:A033451 - cp's OEIS frontend

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].

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

A059044 Initial primes of sets of 5 consecutive primes in arithmetic progression.

Original entry on oeis.org

9843019, 37772429, 53868649, 71427757, 78364549, 79080577, 98150021, 99591433, 104436889, 106457509, 111267419, 121174811, 121174841, 168236119, 199450099, 203908891, 207068803, 216618187, 230952859, 234058871, 235524781, 253412317, 263651161, 268843033, 294485363, 296239787

Offset: 1

Harvey Dubner (harvey(AT)dubner.com), Dec 18 2000

nonn

Each set has a constant difference of 30, for all of the terms listed so far.
It is conjectured that there exist arbitrarily long sequences of consecutive primes in arithmetic progression. As of December 2000, the record is 10 primes.
The first CPAP-5 with common difference 60 starts at 6182296037 ~ 6e9, cf. A210727. This sequence consists of first members of pairs of consecutive primes in A054800 (see also formula): a(1..6) = A054800({1555, 4555, 6123, 7695, 8306, 8371}). Conversely, pairs of consecutive primes in this sequence yield a term of A058362, i.e., they start a sequence of 6 consecutive primes in arithmetic progression (CPAP-6): e.g., the nearby values a(12) = 121174811, a(13) = 121174841 = a(12) + 30 indicate such a term, whence A006560(6) = A058362(1) = a(12). The first CPAP-6 with common difference 60 starts at 293826343073 ~ 3e11, cf. A210727. Longer CPAP's must have common difference >= 210. - M. F. Hasler, Oct 26 2018
About 500 initial terms of this sequence are the same as for the sequence "First of 5 consecutive primes separated by gaps of 30". The first 10^4 terms of A052243 give 281 terms of this sequence (up to ~ 3.34e9) with the same formula as the one using A054800, but as the above comment says, this will miss terms beyond twice that range. - M. F. Hasler, Jan 02 2020

David Wells, The Penguin Dictionary of Curious and Interesting Numbers (Rev. ed. 1997), p. 181.

Zak Seidov, Table of n, a(n) for n = 1..241 (all terms up to 3*10^9)
Jens K. Andersen, The Largest Known CPAP's, updated Sept. 2018
OEIS wiki, Consecutive primes in arithmetic progression, updated Jan. 2020
Index entries for sequences related to primes in arithmetic progressions

Cf. A054800: start of 4 consecutive primes in arithmetic progression (CPAP-4).
Cf. A033451, A033447, A033448, A052242, A052243, A058252, A058323, A067388: start of CPAP-4 with common difference 6, 12, 18, ..., 48.
Cf. A052239: start of first CPAP-4 with common difference 6n.
Cf. A058362: start of 6 consecutive primes in arithmetic progression.
Cf. A006560: first prime to start a CPAP-n.

Select[Partition[Prime[Range[14000000]],5,1],Length[Union[ Differences[ #]]]==1&] (* Harvey P. Dale, Jun 22 2013 *)

A059044(n,p=2,c,g,P)={forprime(q=p+1,, if(p+g!=p+=g=q-p, next, q!=P+2*g, c=3, c++>4, print1(P-2*g,",");n--||break);P=q-g);P-2*g} \\ This does not impose the gap to be 30, but it happens to be the case for the first values. - M. F. Hasler, Oct 26 2018

Found by exhaustive search for 5 primes in arithmetic progression with all other intermediate numbers being composite.

A059044 = { A054800(i) | A054800(i+1) - A151800(A054800(i)) } with the nextprime function A151800(prime(k)) = prime(k+1) = prime(k) + A001223(k). - M. F. Hasler, Oct 27 2018, edited Jan 02 2020.

a(16)-a(22) from Donovan Johnson, Sep 05 2008
Reference added by Harvey P. Dale, Jun 22 2013
Edited (definition clarified, cross-references corrected and extended) by M. F. Hasler, Oct 26 2018

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

A052242 Initial prime in set of 4 consecutive primes in arithmetic progression with common difference 24.

Original entry on oeis.org

1397609, 1436339, 2270459, 4181669, 4231919, 4425599, 4650029, 4967329, 7124099, 8254049, 8431369, 9000379, 9149639, 11343509, 11584009, 11734249, 12867319, 13723009, 14461229, 14590159, 15587659, 15776239, 15932899

Offset: 1

Labos Elemer, Jan 31 2000

nonn

Analogous sequences [with common difference in square brackets]: A033451 [6], A033447 [12], A033448 [18], A052242 [24], A052243 [30].

A052242 = Reap[For[p = 2, p < 16000000, p = NextPrime[p], p2 = NextPrime[p]; If[p2 - p == 24, p3 = NextPrime[p2]; If[p3 - p2 == 24, p4 = NextPrime[p3]; If[p4 - p3 == 24, Print[p]; Sow[p]]]]]][[2, 1]] (* Jean-François Alcover, Jun 28 2012 *)
Transpose[Select[Partition[Prime[Range[1100000]],4,1],Union[ Differences[#]] == {24}&]][[1]] (* Harvey P. Dale, Jun 17 2014 *)

More terms from Harvey P. Dale, Nov 25 2000

Definition clarified by Harvey P. Dale, Jun 17 2014

A058252 Initial prime in set of 4 consecutive primes with common difference 36.

Original entry on oeis.org

5321191, 8606621, 9148351, 41675791, 43251251, 49820291, 51825461, 57791281, 66637721, 73114441, 74055851, 82584841, 86801801, 87620011, 112161451, 123720361, 125810021, 126265751, 136413721, 140969291, 152777291, 153348161

Offset: 1

Harvey P. Dale, Dec 05 2000

nonn

Subsequence of A052197. - R. J. Mathar, Apr 12 2008
There are no 5 consecutive primes with common gap 36. - Zak Seidov, Jan 17 2013
If the primes are not required to be consecutive, the sequence starts 31, 241, 281, 311, 751, 911, 941, 1151, 1451, 2621, 4021, ... - Michael B. Porter, Jan 17 2013

Analogous sequences [with common difference in square brackets]: A033451 [6], A033447 [12], A033448 [18], A052242 [24], A052243 [30]

Transpose[Select[Partition[Prime[Range[8700000]],4,1], Union[ Differences[#]] =={36}&]][[1]]

a(16)-a(22) from Donovan Johnson, Sep 05 2008

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

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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A059044 Initial primes of sets of 5 consecutive primes in arithmetic progression.

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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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A052242 Initial prime in set of 4 consecutive primes in arithmetic progression with common difference 24.

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A058252 Initial prime in set of 4 consecutive primes with common difference 36.

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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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