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
p=67,67+4=71,67+4+2=73,67+4+2+6=79 are consecutive primes.
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].
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d = {4, 2, 6}; First /@ Select[Partition[Prime@ Range@ 12000, Length@ d + 1, 1], Differences@ # == d &] (* Michael De Vlieger, May 02 2016 *)
A078959
Primes p such that the differences between the 5 consecutive primes starting with p are (6,2,6,4).
Original entry on oeis.org
23, 53, 263, 1283, 2333, 5843, 6563, 14543, 19373, 32363, 41603, 48473, 49193, 51413, 75983, 88793, 106853, 113153, 115763, 138563, 150203, 160073, 163973, 204353, 223823, 229763, 246923, 284723, 319673, 326993, 337853, 338153, 357653, 433253, 443153, 460073, 460973
Offset: 1
53 is a term since 53, 59 = 53 + 6, 61 = 53 + 8, 67 = 53 + 14 and 71 = 53 + 18 are consecutive primes.
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l = {}; For[n = 1, n < 10^5, n++, If[Prime[n] + 6 == Prime[n + 1] \[And] Prime[n] + 8 == Prime[n + 2] \[And] Prime[n] + 14 == Prime[n + 3] \[And] Prime[n] + 18 == Prime[n + 4], AppendTo[l, Prime[n]]]]; l (* Jake Foster, Oct 27 2008 *)
Select[Partition[Prime[Range[50000]], 5, 1], Differences[#] == {6,2,6,4} &][[;;, 1]] (* Amiram Eldar, Feb 22 2025 *)
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list(lim) = {my(p1 = 2, p2 = 3, p3 = 5, p4 = 7); forprime(p5 = 11, lim, if(p2 - p1 == 6 && p3 - p2 == 2 && p4 - p3 == 6 && p5 - p4 == 4, print1(p1, ", ")); p1 = p2; p2 = p3; p3 = p4; p4 = p5);} \\ Amiram Eldar, Feb 22 2025
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
p=1997, q=2011 has difference pattern [2,4,8] and {1997,1999,2003,2011} is the corresponding consecutive prime 4-tuple.
A078960
Primes p such that the differences between the 5 consecutive primes starting with p are (6,2,6,6).
Original entry on oeis.org
593, 4643, 6353, 11483, 19463, 34253, 71333, 77543, 89513, 101273, 135593, 148853, 179813, 184823, 191453, 193373, 245513, 260003, 267893, 277883, 280583, 302573, 307253, 308303, 310223, 344243, 346433, 350423, 376463, 408203, 416393, 435563, 442823, 450473, 482393
Offset: 1
593 is in the sequence since 593, 599 = 593 + 6, 601 = 593 + 8, 607 = 593 + 14 and 613 = 593 + 20 are consecutive primes.
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Transpose[Select[Partition[Prime[Range[36000]],5,1],Differences[#]=={6,2,6,6}&]][[1]] (* Harvey P. Dale, Oct 14 2013 *)
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list(lim) = {my(p1 = 2, p2 = 3, p3 = 5, p4 = 7); forprime(p5 = 11, lim, if(p2 - p1 == 6 && p3 - p2 == 2 && p4 - p3 == 6 && p5 - p4 == 6, print1(p1, ", ")); p1 = p2; p2 = p3; p3 = p4; p4 = p5);} \\ Amiram Eldar, Feb 22 2025
A258879
Numbers k such that k is the average of four consecutive primes k-7, k-1, k+1 and k+7.
Original entry on oeis.org
30, 60, 270, 570, 600, 1230, 1290, 1620, 2340, 2550, 3540, 4020, 4650, 5850, 6270, 6360, 6570, 10860, 11490, 14550, 15270, 17490, 19080, 19380, 19470, 23670, 26730, 29130, 32370, 34260, 41610, 48480, 49200, 49530, 51420, 51480
Offset: 1
For k=30: 23, 29, 31, 37 are consecutive primes (k-7=23, k-1=29, k+1=31, k+7=37).
For k=60: 53, 59, 61, 67 are consecutive primes (k-7=53, k-1=59, k+1=61, k+7=67).
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[n: n in [13..2*10^5] | IsPrime(n-7) and IsPrime(n-1) and IsPrime(n+1) and IsPrime(n+7)]; // Vincenzo Librandi Jul 16 2015
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Select[ 5 Range@ 11000, PrimeQ[# - 7] && PrimeQ[# - 1] && PrimeQ[# + 1] && PrimeQ[# + 7] &] (* Robert G. Wilson v, Jun 28 2015 *)
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main(size)={my(v=vector(size),i,t=8);for(i=1,size,while(1,if(isprime(t-7)&&isprime(t-1)&&isprime(t+1)&&isprime(t+7),v[i]=t;break,t++));t++);return(v);} /* Anders Hellström, Jul 17 2015 */
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from sympy import isprime, prevprime, nextprime
for i in range(0, 10001, 2):
if isprime(i-1) and isprime(i+1):
if prevprime(i-1) == i-7 and nextprime(i+1) == i+7: print(i, end=', ')
A282059
Numbers k where there are 8 primes between 10*k and 10*k + 30.
Original entry on oeis.org
1, 8879, 28472, 85571, 114677, 656099, 1576009, 2565844, 6915653, 7426625, 9362599, 18240349, 21803372, 22644952, 26167277, 30254276, 66197230, 91093591, 96466961, 104209078, 107132278, 117022186, 134030186, 139402516, 140053322, 142247591, 145927027
Offset: 1
There are eight primes between 88790 and 88820: 88793, 88799, 88801, 88807, 88811, 88813, 88817, 88819. Therefore 8879 is in the sequence.
For the Dickson conjecture see
A020497.
Comments