A078847 -id:A078847 - cp's OEIS frontend

A078853 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=[6,2,4]; short d-string notation of pattern = [624].

1601, 3911, 5471, 8081, 12101, 12911, 13751, 14621, 17021, 32051, 38321, 40841, 43391, 58901, 65831, 67421, 67751, 68891, 69821, 72161, 80141, 89591, 90011, 90191, 97571, 100511, 102191, 111821, 112241, 122021, 125921, 129281, 129581

Offset: 1

Labos Elemer, Dec 11 2002

nonn

All terms are == 11 (mod 30). Is 180 the minimal first difference? - Zak Seidov, Jun 27 2015

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

			p=1601, 1601+6=1607, 1601+6+2=1609, 1601+6+2+4=1613 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], this sequence[624], A078854[626], A078855[642], A078856[646], A078857[662], A078858[664], A033451[666].

Transpose[Select[Partition[Prime[Range[13000]], 4, 1], Differences[#]=={6, 2, 4} &]][[1]] (* Vincenzo Librandi, Jun 27 2015 *)

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

Listed terms verified by Ray Chandler, Apr 20 2009

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

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

A190814 Initial primes of 5 consecutive primes with consecutive gaps 2, 4, 6, 8.

Original entry on oeis.org

347, 1427, 2687, 4931, 13901, 21557, 23741, 27941, 28277, 31247, 32057, 33617, 45821, 55661, 55817, 68207, 68897, 91571, 128657, 128981, 167621, 179897, 193871, 205421, 221717, 234191, 239231, 258107, 258611, 259157, 278807, 302831, 305477, 348431, 354371

Offset: 1

Zak Seidov, May 20 2011

nonn

All terms = {11,17} mod 30.

a(n) + 20 is the greatest term in the sequence of 5 consecutive primes with 4 consecutive gaps 2, 4, 6, 8. - Muniru A Asiru, Aug 03 2017

			Prime(69..73) = {347, 349, 353, 359, 367} and 349 - 347 = 2, 353 - 349 = 4, 359 - 353 = 6, 367 - 359 = 8.

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

Subsequence of A190799, also subsequence of A078847.

Cf. A190792, A190817, A190819, A190838.

N:= 10^6: # to get all terms <= N
Primes:= select(isprime, [seq(i,i=3..N+20,2)]):
Primes[select(t -> [Primes[t+1]-Primes[t],Primes[t+2]-Primes[t+1],Primes[t+3]-Primes[t+2],Primes[t+4]-Primes[t+3]] = [2,4,6,8], [$1..nops(Primes)-4])]; # Robert Israel, Aug 03 2017

d = Differences[Prime[Range[100000]]]; Prime[Flatten[Position[Partition[d, 4, 1], {2, 4, 6, 8}]]] (* T. D. Noe, May 23 2011 *)
Select[Partition[Prime[Range[31000]],5,1],Differences[#]=={2,4,6,8}&][[All,1]] (* Harvey P. Dale, Jul 03 2020 *)

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

A190817 Initial primes of 6 consecutive primes with consecutive gaps 2,4,6,8,10.

Original entry on oeis.org

13901, 21557, 28277, 55661, 68897, 128981, 221717, 354371, 548831, 665111, 954257, 1164587, 1246367, 1265081, 1538081, 1595051, 1634441, 2200811, 2798921, 2858621, 3053747, 3407081, 3414011, 3967487, 3992201, 4480241, 4608281, 4701731, 4809251, 5029457

Offset: 1

Zak Seidov, May 21 2011

nonn

a(1) = 13901 = A190814(5) = A187058(7) = A078847(24).

a(n) + 30 is the greatest term in the sequence of 6 consecutive primes with consecutive gaps 2, 4, 6, 8, 10. - Muniru A Asiru, Aug 10 2017

			For n = 1, 13901 is in the sequence because 13901, 13903, 13907, 13913, 13921, 13931 are consecutive primes and for n = 2, 21557 is in the sequence since 21557, 21559, 21563, 21569, 21577, 21587 are consecutive primes. - _Muniru A Asiru_, Aug 24 2017

Zak Seidov, Table of n, a(n) for n = 1..6000
R. J. Mathar, Table of Prime Gap Constellations

Cf. A078847, A187058, A190814, A190819, A190838.

K:=3*10^7+1;; # to get all terms <= K.
P:=Filtered([1,3..K],IsPrime);; I:=[2,4,6,8,10];;
P1:=List([1..Length(P)-1],i->P[i+1]-P[i]);;
P2:=List([1..Length(P)-Length(I)],i->[P1[i],P1[i+1],P1[i+2],P1[i+3],P1[i+4]]);;
P3:=List(Positions(P2,I),i->P[i]);  # Muniru A Asiru, Aug 24 2017

N:=10^7: # to get all terms <= N.
Primes:=select(isprime,[seq(i,i=3..N+30,2)]):
Primes[select(t->[Primes[t+1]-Primes[t], Primes[t+2]-Primes[t+1],Primes[t+3]-Primes[t+2], Primes[t+4]-Primes[t+3], Primes[t+5]-Primes[t+4]]=[2,4,6,8,10], [$1..nops(Primes)-5])]; # Muniru A Asiru, Aug 04 2017

d = Differences[Prime[Range[100000]]]; Prime[Flatten[Position[Partition[d, 5, 1], {2, 4, 6, 8, 10}]]] (* T. D. Noe, May 23 2011 *)
With[{s = Differences@ Prime@ Range[10^6]}, Prime[SequencePosition[s, Range[2, 10, 2]][[All, 1]] ] ] (* Michael De Vlieger, Aug 16 2017 *)

lista(nn) = forprime(p=13901, nn, if(nextprime(p+1)==p+2 && nextprime(p+3)==p+6 && nextprime(p+7)==p+12 && nextprime(p+13)==p+20 && nextprime(p+21)==p+30, print1(p", "))); \\ Altug Alkan, Aug 16 2017

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

A190819 Initial primes of 7 consecutive primes with consecutive gaps 2, 4, 6, 8, 10, 12.

Original entry on oeis.org

128981, 665111, 2798921, 3992201, 5071667, 5093507, 5344247, 10732817, 11920367, 16197947, 16462541, 16655447, 16943471, 21456047, 25793897, 32634311, 34051007, 34864211, 35250431, 38585201, 39898757, 49584371, 50375861, 51867197, 54738767, 55793951

Offset: 1

Zak Seidov, May 21 2011

nonn

Subsequence of A190817, a(1) = 128981 = A190817(6).

a(n) + 42 is the greatest term in the sequence of 7 consecutive primes with 6 consecutive gaps 2, 4, 6, 8, 10, 12. - Muniru A Asiru, Aug 10 2017

			Prime(12073..12079) = {128981, 128983, 128987, 128993, 129001, 129011, 129023} with first differences {2, 4, 6, 8, 10, 12}.

Zak Seidov, Table of n, a(n) for n = 1..300

Cf. A078847, A190814, A190817, A190838.

N:=10^7: # to get all terms <= N.
Primes:=select(isprime,[seq(i,i=3..N+42,2)]):
Primes[select(t->[Primes[t+1]-Primes[t], Primes[t+2]-Primes[t+1],
Primes[t+3]-Primes[t+2], Primes[t+4]-Primes[t+3], Primes[t+5]-Primes[t+4], Primes[t+6]-Primes[t+5] ]=[2,4,6,8,10,12], [$1..nops(Primes)-6])]; # Muniru A Asiru, Aug 04 2017

d = Differences[Prime[Range[1000000]]]; Prime[Flatten[Position[Partition[d, 6, 1], {2, 4, 6, 8, 10, 12}]]] (* T. D. Noe, May 23 2011 *)
Prime[SequencePosition[Differences[Prime[Range[34*10^5]]],{2,4,6,8,10,12}][[All,1]]] (* Harvey P. Dale, Feb 18 2022 *)

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

A190838 Initial primes of 8 consecutive primes with 7 consecutive gaps 2, 4, 6, 8, 10, 12, 14.

Original entry on oeis.org

128981, 21456047, 34864211, 51867197, 55793951, 69726647, 113575727, 180078317, 207664397, 232728647, 342241967, 382427027, 382533311, 470463011, 558791327, 591360851, 603413801, 749930717, 838115711, 926976431, 965761397, 1007421251, 1109867567, 1278189947

Offset: 1

Zak Seidov, May 21 2011

nonn

a(1) = 128981 = A190819(1), a(2) = 21456047 = A190819(14).

a(n) + 56 is the greatest term in the sequence of 8 consecutive primes with 7 consecutive gaps 2, 4, 6, 8, 10, 12, 14. - Muniru A Asiru, Aug 10 2017

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Index entries for primes, gaps between

Subsequence of A190819.
Subsequence of A187060. - Michel Marcus, Aug 10 2017
Cf. A078847, A190814, A190817.

N:=10^8:  # to get all terms <= N.
Primes:=select(isprime,[seq(i,i=3..N+56,2)]):
Primes[select(t->[Primes[t+1]-Primes[t], Primes[t+2]-Primes[t+1],
  Primes[t+3]-Primes[t+2], Primes[t+4]-Primes[t+3], Primes[t+5]-
  Primes[t+4], Primes[t+6]-Primes[t+5] , Primes[t+7]-Primes[t+6] ]=
[2,4,6,8,10,12,14], [$1..nops(Primes)-7])]; # Muniru A Asiru, Aug 04 2017

Transpose[Select[Partition[Prime[Range[65000000]],8,1],Differences[#] =={2,4,6,8,10,12,14}&]][[1]] (* Harvey P. Dale, May 10 2014 *)

list(lim)=my(v=List(),p=128981,t); forprime(q=p+2,lim+56, if(q-p-t==2, t+=2; if(t==14, listput(v, q-56); t=0), t=0); p=q); Vec(v) \\ Charles R Greathouse IV, Aug 10 2017

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

A079016 Suppose p and q = p+12 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 14 possible difference patterns, namely [12], [2,10], [4,8], [6,6], [8,4], [10,2], [2,4,6], [2,6,4], [4,2,6], [4,6,2], [6,2,4], [6,4,2], [2,4,2,4] and [4,2,4,2]. Sequence gives smallest value of p for each difference pattern, sorted by magnitude.

Original entry on oeis.org

5, 7, 17, 19, 29, 31, 47, 67, 89, 137, 139, 199, 397, 1601

Offset: 1

Labos Elemer, Jan 24 2003

			p=1601, q=1613 has difference pattern [6,2,4] and {1601,1607,1609,1613} is the corresponding consecutive prime 4-tuple.

A022006(1)=5, A022007(1)=7, A078847(1)=17, A078851(1)=19, A078848(1)=29, A078855(1)=31, A047948(1)=47, A078850(1)=67, A031930(1)=A000230(6)=199, A046137(1)=7, A078853(1)=1601.

Cf. A079017-A079024.

Function[s, Function[t, Union@ Flatten@ Map[s[[First@ Position[t, #]]] &, {{12}, {2, 10}, {4, 8}, {6, 6}, {8, 4}, {10, 2}, {2, 4, 6}, {2, 6, 4}, {4, 2, 6}, {4, 6, 2}, {6, 2, 4}, {6, 4, 2}, {2, 4, 2, 4}, {4, 2, 4, 2}}]]@ Map[Differences@ Select[Range[#, # + 12], PrimeQ] &, s]]@ Select[Prime@ Range[10^3], PrimeQ[# + 12] &] (* Michael De Vlieger, Feb 25 2017 *)

A190792 Primes p=prime(i) such that prime(i+3)-prime(i)=12.

Original entry on oeis.org

17, 19, 29, 31, 41, 59, 61, 67, 71, 127, 227, 229, 269, 271, 347, 431, 607, 641, 1009, 1091, 1277, 1279, 1289, 1291, 1427, 1447, 1487, 1597, 1601, 1607, 1609, 1657, 1777, 1861, 1987, 2129, 2131, 2339, 2371, 2377, 2381, 2539, 2677, 2687, 2707, 2789, 2791

Offset: 1

Zak Seidov, May 20 2011

Minimal distance between prime(i) and prime(i+3) is 12 if all three consecutive prime gaps are different.
There are 6 possible consecutive prime gap configurations:
  {2,4,6}, {2,6,4}, {4,2,6}, {4,6,2}, {6,2,4}, and {6,4,2}.
Least prime quartets with such gap configurations are:
  {17,19,23,29}->{2,4,6}
  {29,31,37,41}->{2,6,4}
  {67,71,73,79}->{4,2,6}
  {19,23,29,31}->{4,6,2}
  {1601,1607,1609,1613}->{6,2,4}
  {31,37,41,43}->{6,4,2}.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A031165, A078847.

[NthPrime(i): i in [2..60000] | NthPrime(i+3)-NthPrime(i) eq 12];  // _Bruno Berselli-, May 20 2011

p = Prime[Range[1000]]; First /@ Select[Partition[p, 4, 1], Last[#] - First[#] == 12 &] (* T. D. Noe, May 23 2011 *)

is(n)=if(!isprime(n), return(0)); my(p=nextprime(n+1),q); if(p-n>6, return(0)); q=nextprime(p+1); q-n<11 && nextprime(q+1)-n==12 \\ Charles R Greathouse IV, Sep 14 2015

A290161 Initial primes of 7 consecutive primes with 6 consecutive gaps 12, 10, 8, 6, 4, 2.

Original entry on oeis.org

752251, 1107751, 4956781, 5647471, 6929401, 10016521, 11516851, 12285631, 18117991, 19280311, 21327961, 21705517, 23946877, 24059011, 24436891, 25976611, 26970751, 29105731, 32254471, 32339521, 32465077, 32542387

Offset: 1

Muniru A Asiru, Jul 22 2017

nonn

All terms = {1,7} mod 30.

For initial primes of 7 consecutive primes with consecutive gaps 2, 4, 6, 8, 10, 12 see A190819.

			Prime(86279..86285) = {1107751, 1107763, 1107773, 1107781, 1107787, 1107791, 1107793 } and 1107751 + 12 = 1107763, 110763 + 10 = 1107773, 1107773 + 8 = 1107781, 1107781 + 6 = 1107787, 1107787 + 4 = 1107791, 1107791 + 2 = 1107793.

Chai Wah Wu, Table of n, a(n) for n = 1..2000

Cf. A078847, A190814, A190817, A190819, A190838, A286891, A290162.

P:=Filtered([1..100000000],IsPrime);; I:=Reversed([2,4,6,8,10,12]);;
P1:=List([1..Length(P)-1],i->P[i+1]-P[i]);;
P2:=List([1..Length(P)-Length(I)],i->[P1[i],P1[i+1],P1[i+2],P1[i+3],P1[i+4],P1[i+5]]);;
P3:=List(Positions(P2,I),i->P[i]);

A290162 Initial primes of 8 consecutive primes with 7 consecutive gaps 14, 12, 10, 8, 6, 4, 2.

Original entry on oeis.org

5647457, 18117977, 21705503, 32465063, 37091597, 57269633, 90217163, 109933673, 111053573, 124123133, 145594583, 146742863, 163123997, 200416343, 239659907, 245333267, 272213813, 335971367, 350795033, 470838833, 701465327, 749927357, 888801707, 1060690667

Offset: 1

Muniru A Asiru, Jul 22 2017

nonn

All terms = {17,23} mod 30.

For initial primes of 8 consecutive primes with consecutive gaps 2, 4, 6, 8, 10, 12, 14 see A190838.

			Prime(390215..390222) = {5647457, 5647471, 5647483, 5647493, 5647501, 5647507, 5647511, 5647513} and 5647457 + 14 = 5647471, 5647471 + 12 = 5647483, 5647483 + 10 = 5647493, 5647493 + 8 = 5647501, 5647501 + 6 = 5647507, 5647507 + 4 = 5647511, 5647511 + 2 = 5647513.

Chai Wah Wu, Table of n, a(n) for n = 1..200

Cf. A078847, A190814, A190817, A190819, A190838, A286891, A290161.

P:=Filtered([1..100000000],IsPrime);; I:=Reversed([2,4,6,8,10,12,14]);;
P1:=List([1..Length(P)-1],i->P[i+1]-P[i]);;  Collected(last);;
P2:=List([1..Length(P)-Length(I)],i->[P1[i],P1[i+1],P1[i+2],P1[i+3],P1[i+4],P1[i+5],P1[i+6]]);;
P3:=List(Positions(P2,I),i->P[i]); Length(P3);

a(8)-a(24) from Giovanni Resta, Jul 25 2017

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A078853 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=[6,2,4]; short d-string notation of pattern = [624].

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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].

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A190814 Initial primes of 5 consecutive primes with consecutive gaps 2, 4, 6, 8.

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A190817 Initial primes of 6 consecutive primes with consecutive gaps 2,4,6,8,10.

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A190819 Initial primes of 7 consecutive primes with consecutive gaps 2, 4, 6, 8, 10, 12.

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A190838 Initial primes of 8 consecutive primes with 7 consecutive gaps 2, 4, 6, 8, 10, 12, 14.

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