A078847 -id:A078847 - cp's OEIS frontend

A078947 Primes p such that the differences between the 5 consecutive primes starting with p are (2,4,6,6).

41, 641, 1091, 4001, 9461, 26681, 26711, 44531, 79811, 103991, 110921, 112571, 172421, 223241, 276821, 289841, 290021, 317771, 373181, 381371, 434921, 450881, 493121, 602081, 678761, 788351, 834131, 907211, 974861, 1076501, 1081121, 1097891, 1200371, 1409531, 1426151

Offset: 1

Labos Elemer, Dec 19 2002

nonn

Equivalently, primes p such that p, p+2, p+6, p+12 and p+18 are consecutive primes.

			641 is in the sequence since 641, 643 = 641 + 2, 647 = 641 + 6, 653 = 641 + 12 and 659 = 641 + 18 are consecutive primes.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Subsequence of A078847.

Cf. A001223, A078866, A078867, A078946-A078971, A022006, A022007.

Select[Partition[Prime[Range[50000]], 5, 1], Differences[#] == {2, 4, 6, 6} &][[;;, 1]] (* Amiram Eldar, Feb 21 2025 *)

list(lim) = {my(p1 = 2, p2 = 3, p3 = 5, p4 = 7); forprime(p5 = 11, lim, if(p2 - p1 == 2 && p3 - p2 == 4 && p4 - p3 == 6 && p5 - p4 == 6, print1(p1, ", ")); p1 = p2; p2 = p3; p3 = p4; p4 = p5);} \\ Amiram Eldar, Feb 21 2025

a(n) == 11 (mod 30). - Amiram Eldar, Feb 21 2025

Edited by Dean Hickerson, Dec 20 2002

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 *)

A079017 Suppose p and q = p+14 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 15 possible difference patterns, namely [14], [2,12], [6,8], [8,6], [12,2], [2,4,8], [2,6,6], [2,10,2], [6,2,6], [6,6,2], [8,4,2], [2,4,6,2], [2,6,4,2], [2,2,4,2,4], [2,4,2,4,2]. Sequence gives smallest value of p for each difference pattern, sorted by magnitude.

Original entry on oeis.org

3, 5, 17, 23, 29, 47, 83, 89, 113, 137, 149, 197, 359, 509, 1997

Offset: 1

Labos Elemer, Jan 24 2003

			p=1997, q=2011 has difference pattern [2,4,8] and {1997,1999,2003,2011} is the corresponding consecutive prime 4-tuple.

A022006(1)=5, A022007(1)=7, A078847(1)=17, A078851(1)=19, A078946(1)=17, A078854(1)=23, A078948(1)=29, A078857(1)=47, A031932(1)=113, A078849(1)=149.

Cf. A079016-A079024.

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

A290706 Greatest of 4 consecutive primes with consecutive gaps 2, 4, 6.

Original entry on oeis.org

29, 53, 239, 359, 653, 1103, 1289, 1439, 1499, 1619, 2699, 3539, 3929, 4013, 4139, 4649, 4799, 4943, 8243, 9473, 10343, 11789, 12119, 13913, 14639, 20759, 21569, 23753, 25589, 26693, 26723, 27749, 27953, 28289, 29033, 31259

Offset: 1

Muniru A Asiru, Aug 09 2017

nonn

All terms = {23, 29} mod 30.

			29 is a member of the sequence because 29 is the greatest of the 4 consecutive primes 17, 19, 23, 29 with consecutive gaps 2, 4, 6.

Cf. A006512, A078847, A098412.

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

for i from 1 to 10^5 do if ithprime(i+1)=ithprime(i)+2 and ithprime(i+2)=ithprime(i)+6 and ithprime(i+3)=ithprime(i)+12 then print(ithprime(i+3)); fi; od;

Select[Prime@ Range@ 3500, NextPrime[#, {1, 2, 3}] == # + {2, 6, 12} &] + 12 (* Giovanni Resta, Aug 09 2017 *)

a(n) = A078847(n) + 12.

cp's OEIS Frontend

A078947 Primes p such that the differences between the 5 consecutive primes starting with p are (2,4,6,6).

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

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

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

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

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A290706 Greatest of 4 consecutive primes with consecutive gaps 2, 4, 6.

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