A055380
Central prime p in the smallest (2n+1)-tuple of consecutive primes that are symmetric with respect to p.
Original entry on oeis.org
5, 18731, 683783, 98303927, 60335249959, 1169769749219, 3945769040699039, 159067808851610657, 6919940122097246597
Offset: 1
In 5-tuple of consecutive primes (18713, 18719, 18731, 18743, 18749), the primes are symmetric w.r.t. its central prime 18731, since 18713+18749 = 18719+18743 = 2*18731, and this is the smallest such 5-tuple. Hence, a(2)=18731.
Alternatively, the symmetry can be seen from the differences between consecutive primes. For (18713, 18719, 18731, 18743, 18749), the differences are (6,12,12,6).
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Table[i = n + 2;
While[x = Differences[Table[Prime[k + i], {k, -n, n}]];
x != Reverse[x], i++]; Prime[i], {n, 3}] (* Robert Price, Oct 12 2019 *)
A055382
Smallest prime starting a sequence of 2n consecutive odd primes with symmetrical gaps about the center.
Original entry on oeis.org
3, 5, 5, 17, 13, 137, 8021749, 1071065111, 1613902553, 1797595814863, 633925574060671, 22930603692243271, 5179852391836338871, 9648166508472058129
Offset: 1
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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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 *)
a(12) from an anonymous participant of the project, added by
Max Alekseyev, Jul 21 2015
A081235
Smallest prime starting a sequence of 2n consecutive primes with symmetrical gaps about the center.
Original entry on oeis.org
2, 5, 5, 17, 13, 137, 8021749, 1071065111, 1613902553, 1797595814863, 633925574060671, 22930603692243271, 5179852391836338871, 9648166508472058129
Offset: 1
The first term is 2 since the 2 primes 2, 3 have a gap of 1, 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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A081235(n) = { my(last=vector(n*=2,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
a(12) from an anonymous participant of the project, added by
Natalia Makarova, Jul 16 2015
A266512
Smallest prime starting a (nonsingular) symmetric n-tuplet of the shortest span (=A266511(n)).
Original entry on oeis.org
2, 3, 47, 5, 18713, 7, 12003179, 17, 1480028129, 13, 1542186111157, 41280160361347, 660287401247633, 10421030292115097, 3112462738414697093, 996689250471604163, 258406392900394343851, 824871967574850703732309, 9425346484752129657862217, 824871967574850703732303
Offset: 1
a(18) from Jaroslaw Wroblewski
A266511
Minimal difference between the smallest and largest of n consecutive large primes that form a symmetric n-tuplet as permitted by divisibility considerations.
Original entry on oeis.org
0, 2, 12, 8, 36, 16, 60, 26, 84, 34, 132, 46, 168, 56, 180, 74, 240, 82, 252, 94, 324, 106, 372, 118, 420, 134, 432, 142, 492, 146, 540, 158, 600, 166, 648, 178, 660, 194, 720, 202, 780, 214, 816, 226, 840, 254, 912, 262, 1020, 278
Offset: 1
For n=3, any shortest symmetric n-tuplet has the form (p, p+6, p+12) and thus a(3)=12.
From _Jon E. Schoenfield_, Jan 05 2016: (Start)
For each n-tuplet (p(1), ..., p(n)) with odd n, let m be its middle prime, i.e., m = p((n+1)/2). Then, since (by symmetry) (p(k) + p(n+1-k))/2 = m for all k = 1..n, we can define the n-tuplet by m and its vector of differences d(j) = m - p(j) for j = 1..(n-1)/2. In other words, given m and d(j) for j = 1..(n-1)/2, the (n-1)/2 primes below m are given by p(j) = m - d(j), and the (n-1)/2 primes above m are given by p(n+1-j) = m + d(j); the difference p(n) - p(1) is thus (m + d(1)) - (m - d(1)) = 2*d(1).
For example, one symmetric 7-tuplet of consecutive primes is (12003179, 12003191, 12003197, 12003209, 12003221, 12003227, 12003239), which can be written as (m-30, m-18, m-12, m, m+12, m+18, m+30) where m=12003209; here we have d(1)=30, d(2)=18, d(3)=12. Among all symmetric 7-tuplets of consecutive primes that satisfy divisibility considerations, the minimal value of d(1) is, in fact, 30, so a(7) = 2*30 = 60.
For n = 3, 5, ..., 29, the lexicographically first vector (d(1), d(2), ..., d((n-1)/2)) permitted by divisibility considerations is as follows:
n| 1 2 3 4 5 6 7 8 9 10 11 12 13 14
--+-------------------------------------------------------
3| 6
5| 18 12
7| 30 18 12
9| 42 30 18 12
11| 66 60 36 24 6
13| 84 66 60 36 24 6
15| 90 84 66 60 36 24 6
17|120 108 90 78 60 48 42 18
19|126 120 114 96 84 54 36 30 6
21|162 150 132 120 108 102 78 48 42 18
23|186 180 150 144 126 96 84 66 60 54 30
25|210 186 180 150 144 126 96 84 66 60 54 30
27|216 210 204 180 126 120 114 96 84 54 36 30 6
29|246 216 210 204 186 174 144 126 90 84 66 60 24 6
(End)
a(15) and a(17)-a(18) from Jaroslaw Wroblewski
a(19), a(21), a(23), a(25), a(27), a(29) from
Jon E. Schoenfield, Jan 02 2016, Jan 05 2016
A266583
Smallest prime starting a symmetric n-tuple of consecutive primes of the smallest span (=A266676(n)).
Original entry on oeis.org
2, 2, 3, 5, 18713, 5, 12003179, 17, 1480028129, 13, 1542186111157, 41280160361347, 660287401247633, 10421030292115097, 3112462738414697093, 996689250471604163, 258406392900394343851, 824871967574850703732309, 9425346484752129657862217, 824871967574850703732303
Offset: 1
A336967
Prime starting a sequence of 24 consecutive primes with symmetrical gaps about the center.
Original entry on oeis.org
22930603692243271, 34984922852185283, 60960572612579749, 226721453950385059, 301850075265898823, 310402815525745511, 341206644560627711, 357582484287837103, 481408770994035947, 492720459594614777, 528050771271601307, 587950582712698157, 675424273001524577
Offset: 1
Cf.
A000040,
A055381,
A055382,
A064101,
A081235,
A175309,
A335044,
A335394,
A336966,
A336968,
A359440.
A336968
Prime starting a sequence of 22 consecutive primes with symmetrical gaps about the center.
Original entry on oeis.org
633925574060671, 2235053194261739, 3693434256575461, 6244996197964523, 7312449941282693, 11768508587048027, 12241378636561883, 12696156429346387, 13388148635660387, 14052415423668901, 18620445306703861, 19802687937976219, 22930603692243341, 23122811970297833
Offset: 1
Cf.
A000040,
A055381,
A055382,
A064101,
A081235,
A175309,
A333977,
A335044,
A335394,
A336967,
A359440.
A266585
Smallest m such that prime(m) starts a symmetric n-tuple of consecutive primes of the smallest span (=A266676(n)).
Original entry on oeis.org
1, 1, 2, 3, 2136, 3, 788244, 7, 73780392, 6, 57067140928, 1361665032086, 19953429852608, 290660101635794, 74896929428416952, 24660071077535201, 5620182896687887031
Offset: 1
A266676
Smallest span (difference between the start and end) of a symmetric n-tuple of consecutive primes.
Original entry on oeis.org
0, 1, 4, 8, 36, 14, 60, 26, 84, 34, 132, 46, 168, 56, 180, 74, 240, 82
Offset: 1
The smallest starting primes and their indices of the corresponding tuples are given in
A266583 and
A266585.
Showing 1-10 of 13 results.
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