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Search: a081235 -id:a081235
Displaying 1-10 of 11 results found. page 1 2
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A055382 Smallest prime starting a sequence of 2n consecutive odd primes with symmetrical gaps about the center. +10
14
3, 5, 5, 17, 13, 137, 8021749, 1071065111, 1613902553, 1797595814863, 633925574060671, 22930603692243271 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
a(13) <= 5179852391836338871. The solution was found by the BOINC project "SPT test project". - Natalia Makarova, Dec 06 2023
LINKS
Table of n, a(n) for n=1..12.
Discussion at the scientific forum dxdy.ru, Distributed computing project (in Russian)
Symmetric Prime Tuples, SPT test project
Index entries for primes, gaps between
FORMULA
For n>1, a(n) = A081235(n) = A175309(2n-1).
EXAMPLE
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.
MATHEMATICA
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 *)
CROSSREFS
Cf. A001223, A055380, A055381, A175309.
See A081235 for another version.
KEYWORD
more,nonn
AUTHOR
Jud McCranie, Jun 23 2000
EXTENSIONS
a(10) from Donovan Johnson, Mar 09 2008
Minor edits by N. J. A. Sloane, Apr 02 2010
a(11) from Dmitry Petukhov, added by Max Alekseyev, Aug 08 2014
a(12) from an anonymous participant of the project, added by Max Alekseyev, Jul 21 2015
STATUS
approved
A175309 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. +10
14
2, 3, 5, 18713, 5, 683747, 17, 98303867, 13, 60335249851, 137, 1169769749111, 8021749, 3945769040698829, 1071065111, 159067808851610411, 1613902553 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
From M. F. Hasler, Apr 02 2010: (Start)
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
LINKS
Table of n, a(n) for n = 1..17
BOINC project to search all up to 2^64
FORMULA
a(2n-1) = A081235(n) (= A055382(n) for n>1). - M. F. Hasler, Apr 02 2010
MATHEMATICA
A175309[n_] := Module[{k},
k = 1; While[! AllTrue[Range[n], Prime[k+#] - Prime[k+#-1] ==
Prime[n+k+1-#] - Prime[n+k-#] &], k++]; Return[Prime[k]]];
Table[A175309[n], {n, 1, 7}] (* Robert Price, Mar 27 2019 *)
PROG
(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
CROSSREFS
Cf. A006562, A051795, A081235, A081415, A096710, A055382.
KEYWORD
nonn,more
AUTHOR
Leroy Quet, Mar 27 2010
EXTENSIONS
Terms through a(12) were calculated by (in alphabetical order) Franklin T. Adams-Watters, Hans Havermann and D. S. McNeil
Minor edits by N. J. A. Sloane, Apr 02 2010
a(14) from Dmitry Petukhov, added by Max Alekseyev, Nov 03 2014
a(16) from BOINC project, added by Dmitry Petukhov, Apr 06 2017
STATUS
approved
A359440 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. +10
10
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,5
COMMENTS
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
LINKS
Table of n, a(n) for n=1..87.
FORMULA
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
EXAMPLE
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.
PROG
(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)
CROSSREFS
Cf. A000040, A006562, A051795, A055381, A081235.
KEYWORD
nonn
AUTHOR
Alexandre Herrera, Jan 01 2023
EXTENSIONS
Introductory phrase added to name by Peter Munn, Jan 08 2023
STATUS
approved
A335044 Primes starting 14-tuples of consecutive primes that have symmetrical gaps about their mean and form 7 pairs of twin primes. +10
5
1855418882807417, 2485390773085247, 4038284355308309, 14953912258447817, 16152884167551797, 20149877129714999, 23535061700758967, 24067519779525107, 25892136591156917, 28681238268465371, 29359755788438639, 38364690814563809, 52367733685120277 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Tomáš Brada, Table of n, a(n) for n = 1..122
EXAMPLE
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.
CROSSREFS
Cf. A330278, A274792, A055380, A055381, A055382, A081235, A266511, A266512.
KEYWORD
nonn
AUTHOR
Tomáš Brada, Jun 05 2020
STATUS
approved
A335394 Primes starting 16-tuples of consecutive primes that have symmetrical gaps about their mean and form 8 pairs of twin primes. +10
4
2640138520272677, 119890755200639999, 156961225134536189, 193609877401516181, 215315384130681929, 404072710417411769, 517426190585100089, 519460320704755811 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Table of n, a(n) for n=1..8.
EXAMPLE
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.
CROSSREFS
Cf. A330278, A274792, A055380, A055381, A055382, A081235, A266511, A266512.
KEYWORD
nonn,more
AUTHOR
Tomáš Brada, Natalia Makarova Jun 05 2020
STATUS
approved
A336967 Prime starting a sequence of 24 consecutive primes with symmetrical gaps about the center. +10
4
22930603692243271, 34984922852185283, 60960572612579749, 226721453950385059, 301850075265898823, 310402815525745511, 341206644560627711, 357582484287837103, 481408770994035947, 492720459594614777, 528050771271601307, 587950582712698157, 675424273001524577 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Tomáš Brada, Table of n, a(n) for n = 1..18
Tomáš Brada, Natalia Makarova, Symmetric Prime Tuples project
Natalia Makarova, About Stop@home project
Natalia Makarova and Carlos Rivera, Problem 62. Symmetric k-tuples of consecutive primes, The Prime Puzzles and Problems Connection.
FORMULA
Primes p = prime(k) = A000040(k) such that A359440(k+11) >= 11. - Peter Munn, Jan 09 2023
CROSSREFS
Cf. A000040, A055381, A055382, A064101, A081235, A175309, A335044, A335394, A336966, A336968, A359440.
KEYWORD
nonn
AUTHOR
Tomáš Brada, Aug 09 2020
STATUS
approved
A336968 Prime starting a sequence of 22 consecutive primes with symmetrical gaps about the center. +10
4
633925574060671, 2235053194261739, 3693434256575461, 6244996197964523, 7312449941282693, 11768508587048027, 12241378636561883, 12696156429346387, 13388148635660387, 14052415423668901, 18620445306703861, 19802687937976219, 22930603692243341, 23122811970297833 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Tomáš Brada, Table of n, a(n) for n = 1..345
Tomáš Brada, Natalia Makarova, Symmetric Prime Tuples project
Natalia Makarova, About Stop@home project
Natalia Makarova and Carlos Rivera, Problem 62. Symmetric k-tuples of consecutive primes, The Prime Puzzles and Problems Connection.
FORMULA
Primes p = prime(k) = A000040(k) such that A359440(k+10) >= 10. - Peter Munn, Jan 09 2023
CROSSREFS
Cf. A000040, A055381, A055382, A064101, A081235, A175309, A333977, A335044, A335394, A336967, A359440.
KEYWORD
nonn
AUTHOR
Tomáš Brada, Aug 09 2020
STATUS
approved
A330278 Primes starting 12-tuples of consecutive primes that have symmetrical gaps about their mean and form 6 pairs of twin primes. +10
3
17479880417, 158074620437, 1071796554401, 1087779101699, 1153782400787, 1628444511389, 2066102452949, 2083857437327, 2561560206377, 3731086236287, 3751571181929, 4158362831639, 4878193583477, 5008751356547, 5378606656847, 5531533689527, 7020090738707, 7036216236989 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Tomáš Brada, Table of n, a(n) for n = 1..10000 (terms a(7)-a(132) from Giovanni Resta, terms a(133)-a(228) from Franz-Xaver Harvanek)
EXAMPLE
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.
CROSSREFS
Cf. A055380, A055381, A055382, A081235, A266511, A266512.
KEYWORD
nonn
AUTHOR
Max Alekseyev, Dec 08 2019
EXTENSIONS
a(2)-a(6) from Franz-Xaver Harvanek
More terms from Giovanni Resta, Dec 10 2019
STATUS
approved
A336966 Primes starting 10-tuples of consecutive primes that have symmetrical gaps about their mean and form 5 pairs of twin primes. +10
2
3031329797, 5188151387, 14168924459, 14768184029, 18028534367, 26697800819, 26919220961, 29205326387, 32544026699, 39713433671, 45898528799, 48263504459, 50791655009, 66419473031, 71525244611, 80179195037, 83700877199, 86767580069, 97660776137, 108116163479 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Table of n, a(n) for n=1..20.
EXAMPLE
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.
CROSSREFS
Cf. A055380, A055381, A055382, A081235, A266511, A266512, A335044, A335394, A336966, A336968.
KEYWORD
nonn
AUTHOR
Tomáš Brada, Aug 09 2020
STATUS
approved
A333977 Prime starting a sequence of 20 consecutive primes with symmetrical gaps about the center. +10
1
1797595814863, 2375065608481, 4465545586753, 21818228348093, 67696772430071, 82116093014611, 155947272322087, 161980267642951, 169560139541641, 202619277419161, 285719200081877, 299828814652799, 314942862282899, 365706921997577 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Table of n, a(n) for n=1..14.
Tomáš Brada, Natalia Makarova, Symmetric Prime Tuples project
Natalia Makarova, About Stop@home project
Natalia Makarova and Carlos Rivera, Problem 62. Symmetric k-tuples of consecutive primes, The Prime Puzzles and Problems Connection.
FORMULA
Primes p = prime(k) = A000040(k) such that A359440(k+9) >= 9. - Peter Munn, Jan 09 2023
CROSSREFS
Cf. A000040, A055381, A055382, A064101, A081235, A175309, A335044, A335394, A336967, A336968, A359440.
KEYWORD
nonn
AUTHOR
Tomáš Brada, Sep 20 2020
STATUS
approved
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Last modified September 11 17:54 EDT 2024. Contains 375839 sequences. (Running on oeis4.)