Search: a081235 -id:a081235
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A055382
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Smallest prime starting a sequence of 2n consecutive odd primes with symmetrical gaps about the center.
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+10
14
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3, 5, 5, 17, 13, 137, 8021749, 1071065111, 1613902553, 1797595814863, 633925574060671, 22930603692243271
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OFFSET
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1,1
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COMMENTS
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a(13) <= 5179852391836338871. The solution was found by the BOINC project "SPT test project". - Natalia Makarova, Dec 06 2023
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LINKS
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FORMULA
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EXAMPLE
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The first term is 3 since the 2 primes 3, 5 have a gap of 2, which is trivially symmetric about its center.
The second term is 5 since the 4 primes 5, 7, 11, 13 have gaps 2, 4, 2, which is symmetric about its center.
The twelve primes 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193 have gaps 2, 10, 2, 6, 6, 4, 6, 6, 2, 10, 2 - symmetric about the middle, so a(6) = 137.
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MATHEMATICA
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Table[i = 1;
While[x = Differences[Table[Prime[k + i], {k, 2 n}]];
x != Reverse[x], i++]; Prime[i + 1], {n, 6}] (* Robert Price, Oct 12 2019 *)
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CROSSREFS
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KEYWORD
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more,nonn
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AUTHOR
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EXTENSIONS
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a(12) from an anonymous participant of the project, added by Max Alekseyev, Jul 21 2015
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STATUS
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approved
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A175309
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a(n) = the smallest prime prime(k) such that prime(k+j) - prime(k+j-1) = prime(n+k+1-j) - prime(n+k-j) for all j with 1 <= j <= n.
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+10
14
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2, 3, 5, 18713, 5, 683747, 17, 98303867, 13, 60335249851, 137, 1169769749111, 8021749, 3945769040698829, 1071065111, 159067808851610411, 1613902553
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OFFSET
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1,1
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COMMENTS
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Also: Start of the first sequence of n+1 consecutive primes symmetrically distributed w.r.t. their barycenter, e.g., [2,3], [3,5,7], [5,7,11,13], [18713, 18719, 18731, 18743, 18749]. With this definition, it would make sense to prefix the sequence with an initial term a(0)=2.
Sequence A081235 (or A055382, which is essentially the same) consists of every other term of this sequence. (End)
a(19) = 1797595814863, a(21) = 633925574060671, a(23) = 22930603692243271. - Tomáš Brada, May 25 2020
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LINKS
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FORMULA
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MATHEMATICA
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k = 1; While[! AllTrue[Range[n], Prime[k+#] - Prime[k+#-1] ==
Prime[n+k+1-#] - Prime[n+k-#] &], k++]; Return[Prime[k]]];
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PROG
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(PARI) a(n)={ my( last=vector(n++, i, prime(i)), m, i=Mod(n-2, n)); forprime(p=last[n], default(primelimit), m=last[1+lift(i+2)]+last[1+lift(i++)]=p; for( j=1, n\2, last[1+lift(i-j)]+last[1+lift(i+j+1)]==m || next(2)); return( last[1+lift(i+1)])) } \\ M. F. Hasler, Apr 02 2010
(PARI) isok(p, n) = {my(k=primepi(p)); for (j=1, n, if (prime(k+j) - prime(k+j-1) != prime(n+k+1-j) - prime(n+k-j), return (0)); ); return (1); } \\ Michel Marcus, Apr 08 2017
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A359440
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A measure of the extent of reflective symmetry in the pattern of primes around each prime gap: a(n) is the largest k such that prime(n-j) + prime(n+1+j) has the same value for each j in 0..k.
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+10
10
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0, 0, 0, 1, 2, 2, 1, 0, 0, 4, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 2, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 5, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0
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OFFSET
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1,5
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COMMENTS
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If the prime gaps above and below a prime p have the same length, p is called a balanced prime (see A006562). Likewise, if the prime gaps above and below the n-th prime gap have the same length, this gap might be called a balanced prime gap. These gaps correspond to nonzero terms a(n). Similarly, if a(n) >= 2, the n-th prime gap is the equivalent of a doubly balanced prime (A051795), and so on. - Peter Munn, Jan 08 2023
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LINKS
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FORMULA
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a(n) = min( {n-1} U {k : 0 <= k <= n-2 and prime(n-k-1) + prime(n+k+2) <> prime(n) + prime(n+1)} ). - Peter Munn, Jan 08 2023
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EXAMPLE
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For n = 1, prime(1) + prime(2) = 2 + 3 = 5; "prime(0)" does not exist, so a(1) = 0.
For n = 4:
j = 0: prime(4) + prime(5) = 7 + 11 = 18;
j = 1: prime(3) + prime(6) = 5 + 13 = 18;
j = 2: prime(2) + prime(7) = 3 + 17 = 20 != 18, so a(4) = 1.
For n = 5:
j = 0: prime(5) + prime(6) = 11 + 13 = 24;
j = 1: prime(4) + prime(7) = 7 + 17 = 24;
j = 2: prime(3) + prime(8) = 5 + 19 = 24;
j = 3: prime(2) + prime(9) = 3 + 23 = 26 != 24, so a(5) = 2.
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PROG
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(Python)
import sympy
offset = 1
N = 100
l = []
for n in range(offset, N+1):
j = 0
first_sum = sympy.prime(n-j)+sympy.prime(n+j+1)
while (n-j) > 1:
j += 1
sum = sympy.prime(n-j)+sympy.prime(n+j+1)
if sum != first_sum:
break
l.append(max(0, j-1))
print(l)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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Introductory phrase added to name by Peter Munn, Jan 08 2023
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STATUS
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approved
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A335044
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Primes starting 14-tuples of consecutive primes that have symmetrical gaps about their mean and form 7 pairs of twin primes.
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+10
5
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1855418882807417, 2485390773085247, 4038284355308309, 14953912258447817, 16152884167551797, 20149877129714999, 23535061700758967, 24067519779525107, 25892136591156917, 28681238268465371, 29359755788438639, 38364690814563809, 52367733685120277
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OFFSET
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1,1
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LINKS
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EXAMPLE
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a(1) = A274792(7) = 1855418882807417 starts a 14-tuple of consecutive primes: 1855418882807417+s for s in {0 2 12 14 30 32 72 74 114 116 132 134 144 146} that are symmetric about 1855418882807417+73 and form 7 pairs of twin primes.
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A335394
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Primes starting 16-tuples of consecutive primes that have symmetrical gaps about their mean and form 8 pairs of twin primes.
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+10
4
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OFFSET
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1,1
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LINKS
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EXAMPLE
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a(1) = A274792(8) = 2640138520272677 starts a 16-tuple of consecutive primes: 2640138520272677+s for s in {0, 2, 12, 14, 30, 32, 54, 56, 90, 92, 114, 116, 132, 134, 144, 146} that are symmetric about 2640138520272677+73 and form 8 pairs of twin primes.
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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A336967
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Prime starting a sequence of 24 consecutive primes with symmetrical gaps about the center.
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+10
4
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22930603692243271, 34984922852185283, 60960572612579749, 226721453950385059, 301850075265898823, 310402815525745511, 341206644560627711, 357582484287837103, 481408770994035947, 492720459594614777, 528050771271601307, 587950582712698157, 675424273001524577
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OFFSET
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1,1
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LINKS
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FORMULA
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CROSSREFS
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Cf. A000040, A055381, A055382, A064101, A081235, A175309, A335044, A335394, A336966, A336968, A359440.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A336968
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Prime starting a sequence of 22 consecutive primes with symmetrical gaps about the center.
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+10
4
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633925574060671, 2235053194261739, 3693434256575461, 6244996197964523, 7312449941282693, 11768508587048027, 12241378636561883, 12696156429346387, 13388148635660387, 14052415423668901, 18620445306703861, 19802687937976219, 22930603692243341, 23122811970297833
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OFFSET
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1,1
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LINKS
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FORMULA
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CROSSREFS
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Cf. A000040, A055381, A055382, A064101, A081235, A175309, A333977, A335044, A335394, A336967, A359440.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A330278
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Primes starting 12-tuples of consecutive primes that have symmetrical gaps about their mean and form 6 pairs of twin primes.
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+10
3
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17479880417, 158074620437, 1071796554401, 1087779101699, 1153782400787, 1628444511389, 2066102452949, 2083857437327, 2561560206377, 3731086236287, 3751571181929, 4158362831639, 4878193583477, 5008751356547, 5378606656847, 5531533689527, 7020090738707, 7036216236989
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OFFSET
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1,1
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LINKS
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EXAMPLE
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a(1) = A274792(6) = 17479880417 starts a 12-tuple of consecutive primes: 17479880417+s for s in {0, 2, 24, 26, 30, 32, 54, 56, 60, 62, 84, 86} that are symmetric about 17479880417+43 and form 6 pairs of twin primes.
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A336966
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Primes starting 10-tuples of consecutive primes that have symmetrical gaps about their mean and form 5 pairs of twin primes.
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+10
2
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3031329797, 5188151387, 14168924459, 14768184029, 18028534367, 26697800819, 26919220961, 29205326387, 32544026699, 39713433671, 45898528799, 48263504459, 50791655009, 66419473031, 71525244611, 80179195037, 83700877199, 86767580069, 97660776137, 108116163479
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OFFSET
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1,1
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LINKS
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EXAMPLE
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a(1) = A274792(5) = 3031329797 starts a 10-tuple of consecutive primes: 3031329797+s for s in {0, 2, 12, 14, 42, 44, 72, 74, 84, 86} that are symmetric about 3031329797+43 and form 5 pairs of twin primes.
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A333977
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Prime starting a sequence of 20 consecutive primes with symmetrical gaps about the center.
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+10
1
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1797595814863, 2375065608481, 4465545586753, 21818228348093, 67696772430071, 82116093014611, 155947272322087, 161980267642951, 169560139541641, 202619277419161, 285719200081877, 299828814652799, 314942862282899, 365706921997577
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OFFSET
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1,1
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LINKS
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FORMULA
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CROSSREFS
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Cf. A000040, A055381, A055382, A064101, A081235, A175309, A335044, A335394, A336967, A336968, A359440.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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