A190838 -id:A190838 - cp's OEIS frontend

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.

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

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

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

A286891 Initial primes of 6 consecutive primes with 5 consecutive gaps 10, 8, 6, 4, 2.

Original entry on oeis.org

41203, 556243, 576193, 715849, 752263, 859249, 891799, 1107763, 1191079, 1201999, 1210369, 1510189, 1601599, 1893163, 2530963, 2678719, 2881243, 3257689, 3431479, 3545263, 3792949, 3804919, 4041109, 4479463, 4867309

Offset: 1

Muniru A Asiru, Jul 22 2017

nonn

All terms = {13,19} mod 30.

For initial primes of 6 consecutive primes with consecutive gaps 2, 4, 6, 8, 10 see A190817.

			Prime(4313..4318) = {41203, 41213, 41221, 41227, 41231, 41233} and 41203 + 10 = 41213, 41213 + 8 = 41221, 41221 + 6 = 41227, 41227 + 4 = 41231, 41231 + 2 = 41233.
Also, prime(68287..68292) = {859249, 859259, 859267, 859273, 859277, 859279} and 859249 + 10 = 859259, 859259 + 8 = 859267, 859267 + 6 = 859273, 859273 + 4 = 859277, 859277 + 2 = 859279.

Chai Wah Wu, Table of n, a(n) for n = 1..10000
R. J. Mathar, Table of Prime Gap Constellations

Cf. A078847, A190814, A190817, A190819, A190838, A290161, A290162.

Subsequence of A031928.

P:=Filtered([1..20000000],IsPrime);;  I:=Reversed([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]);

K:=10^7: # to get all terms <= K.
Primes:=select(isprime,[seq(i,i=3..K+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]]=[10,8,6,4,2], [$1..nops(Primes)-5])]; # Muniru A Asiru, Aug 15 2017

Select[Partition[Prime[Range[340000]],6,1],Differences[#]=={10,8,6,4,2}&][[All,1]] (* Harvey P. Dale, Aug 22 2018 *)

A281448 Initial primes of 9 consecutive primes with consecutive gaps 2, 4, 6, 8, 10, 12, 14, 16.

Original entry on oeis.org

113575727, 232728647, 2426256797, 2469604721, 3344410367, 4656098957, 4952808461, 6369321857, 6430890287, 6760087151, 8518049207, 10818813737, 13195845317, 19684555061, 21884908931, 23953276661, 25509639137, 26367829331, 26390212061, 31004257211

Offset: 1

Harvey P. Dale, Jan 27 2017

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..174

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

Select[Partition[Prime[Range[20*10^7]], 9, 1], Differences[#] == 2*Range[8] &][[All, 1]]

g=0; p=2; forprime(q=3,, if(q-p==g+2, g+=2; if(g==16, print1(q-72", "); g=0), g=0); p=q) \\ Charles R Greathouse IV, Jan 28 2017

a(10)-a(20) from Charles R Greathouse IV, Jan 28 2017

A289907 Initial primes of 5 consecutive primes with consecutive gaps 8,6,4,2.

Original entry on oeis.org

1979, 5399, 11813, 41213, 42443, 44249, 47129, 55799, 57773, 74699, 79613, 84299, 88643, 126473, 143813, 148913, 167099, 176489, 178799, 178889, 209249, 211859, 237143, 266663, 267629, 272249, 272333, 322229, 344153, 348443, 354023, 375083, 391379, 399263, 422069, 449549, 521519, 529673

Offset: 1

Muniru A Asiru, Jul 14 2017

nonn

All terms = {23, 29} mod 30.
For initial primes of 5 consecutive primes with consecutive gaps 2,4,6,8 see A190814.
Number of terms less than 10^k: 0, 0, 0, 2, 13, 65, 317, 1563, 8671, 50643, ..., . - Robert G. Wilson v, Dec 07 2017

			Prime(299..303) = { 1979, 1987, 1993, 1997, 1999 } and 1979 + 8 = 1987, 1987 + 6 = 1993, 1993 + 4 = 1997, 1997 + 2 = 1999.
Also, prime(5852..5856) = { 57773, 57781, 57787, 57791, 57793 } and 5773 + 8 = 57781, 57781 + 6 = 57787, 57787 + 4 = 57791, 57791 + 2 = 57793.

Robert G. Wilson v, Table of n, a(n) for n = 1..10000 (first 3114 terms from Muniru A Asiru)

Cf. A078847, A153419, A190814, A190817, A190819, A190838.

I:=[8,6,4,2];;
P:=Filtered([1..1000000],IsPrime);;
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]]);;
P3:=List(Positions(P2,I),i->P[i]);

s = Prepend[Differences@ #, First@ #] & /@ Partition[Prime@ Range[10^5], 5, 1]; Select[s, Drop[#, 1] == Range[8, 2, -2] &][[All, 1]] (* Michael De Vlieger, Jul 14 2017 *)
p = {2, 3, 5, 7, 11}; lst = {}; While[ p[[1]] < 530000, If[ Differences@ p == {8, 6, 4, 2}, AppendTo[ lst, p[[1]] ]]; p = Join[Rest@ p, {NextPrime[ p[[-1]]] }]]; lst (* Robert G. Wilson v, Dec 07 2017 *)

is(n) = my(q); forstep(i=8,2,-2,q=nextprime(n+1); if(q-n!=i,return(0)); n=q); return(1) \\ David A. Corneth, Jul 23 2017

A290264 Initial primes of 9 consecutive primes with 8 consecutive gaps 16, 14, 12, 10, 8, 6, 4, 2.

Original entry on oeis.org

32465047, 37091581, 146742847, 239659891, 245333251, 272213797, 1060690651, 1541736811, 2002738207, 2480351677, 2636566351, 4421955007, 6168859201, 8158683037, 10367633527, 10623394321, 11452116817, 11691059641, 11892876841, 13551877831, 15043908637

Offset: 1

Muniru A Asiru, Jul 25 2017

nonn

All terms = {1,7} mod 30.
For initial primes of 7 consecutive primes with 6 consecutive gaps 12, 10, 8, 6, 4, 2 and 8 consecutive primes with 7 consecutive gaps 14, 12, 10, 8, 6, 4, 2 see A290161 and A290162 respectively.
a(6) > 250000000.

			32465047 is a member of this sequence because the 9 consecutive primes 32465047, 32465063, 32465077, 32465089, 32465099, 32465107, 32465113, 32465117, 32465119 have consecutive gaps 16, 14, 12, 10, 8, 6, 4, 2. That is, 32465047 + 16 = 32465063, 32465063 + 14 = 32465077, 32465077 + 12 = 32465089, 32465089 + 10 = 32465099, 32465099 + 8 = 32465107, 32465107 + 6 = 32465113, 32465113 + 4 = 32465117, 32465117 + 2 = 32465119.

Cf. A078847, A190814, A190817, A190819, A190838, A290161, A290162.

P:=Filtered([1..50000000],IsPrime);;I:=Reversed([2,4,6,8,10,12,14,16]);;
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],P1[i+7]]);;
P3:=List(Positions(P2,I),i->P[i]); Length(P3);

a(6)-a(21) from Giovanni Resta, Jul 25 2017

cp's OEIS Frontend

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

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

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A286891 Initial primes of 6 consecutive primes with 5 consecutive gaps 10, 8, 6, 4, 2.

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