cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A052160 Isolated prime difference equals 6: primes prime(k) such that d(k) = prime(k+1) - prime(k) = 6 but neither d(k+1) nor d(k-1) is 6.

Original entry on oeis.org

23, 31, 61, 73, 83, 131, 233, 271, 331, 353, 383, 433, 443, 503, 541, 571, 677, 751, 991, 1013, 1033, 1063, 1231, 1283, 1291, 1321, 1433, 1453, 1493, 1543, 1553, 1601, 1613, 1621, 1657, 1777, 1861, 1973, 1987, 2011, 2063, 2131, 2207, 2333, 2341, 2351
Offset: 1

Views

Author

Labos Elemer, Jan 25 2000

Keywords

Comments

Consecutive primes 17, 19, 23, 29, 31 give the pattern of first differences 2, 4, 6, 2 in which the neighboring differences of 6 are not equal to 6.
a(n) - 6 can be prime but not the prime immediately previous to a(n); e.g., 23 - 6 = 17, but the prime 19 lies between the two primes 17 and 23.

Links

Crossrefs

Cf. A001223, A033451, A047948.

Programs

  • Maple
    N:= 3000: # to get all terms <= N
Primes:= select(isprime, [seq(i,i=3..N,2)]):
d:= Primes[2..-1]-Primes[1..-2]:
R:= select(t -> d[t] = 6 and d[t+1] <> 6 and d[t-1] <> 6, [$2..nops(d)-1]):
Primes[R]; # Robert Israel, May 29 2018
  • PARI
    lista(nn) = {vp = primes(nn); vd = vector(#vp-1, k, vp[k+1] - vp[k]); for (i=2, #vd, if ((vd[i] == 6) && (vd[i-1] !=6) && (vd[i+1] != 6), print1(vp[i], ", ")););} \\ Michel Marcus, May 29 2018

A052190 Primes p such that p, p+24, p+48 are consecutive primes.

Original entry on oeis.org

16763, 40039, 42509, 96353, 98573, 104183, 119243, 123863, 160093, 161783, 169259, 181789, 185243, 208529, 209719, 232753, 235699, 243343, 246049, 260339, 261799, 270073, 295363, 295703, 302459, 315199, 331399, 362003, 364079, 373669, 380729, 381793, 385943, 414809
Offset: 1

Views

Author

Labos Elemer, Jan 28 2000

Keywords

Comments

Old name was "Primes p(k) such that p(k+2)-p(k+1)=p(k+1)-p(k)=24."

Examples

			40039 is followed by 40063 and 40087, consecutive primes with equal distance of 24.

Links

Crossrefs

Subsequence of A098974.
Cf. A001223, A033451, A047948, A052188, A052189.

Programs

  • Mathematica
    Select[Partition[Prime[Range[40000]],3,1],Differences[#]=={24,24}&][[All,1]] (* Harvey P. Dale, May 09 2019 *)
  • PARI
    list(lim) = {my(p1 = 2, p2 = 3); forprime(p3 = 5, lim, if(p2 - p1 == 24 && p3 - p2 == 24, print1(p1, ", ")); p1 = p2; p2 = p3);} \\ Amiram Eldar, Feb 28 2025

Extensions

Name changed by Jon E. Schoenfield, May 30 2018

A052197 Primes p such that p, p+36, p+72 are consecutive primes.

Original entry on oeis.org

255767, 704321, 806821, 884501, 913067, 1065137, 1216177, 1448497, 1526191, 1532471, 1640971, 1918571, 2071087, 2275067, 2276431, 2336671, 2347591, 2376721, 2778547, 3098561, 3190601, 3248941, 3259001, 3452107, 3558481
Offset: 1

Views

Author

Labos Elemer, Jan 28 2000

Keywords

Comments

Old name was: Primes p(k) such that p(k+2)-p(k+1)=p(k+1)-p(k)=36.

Examples

			a(3) = 704321 is followed by 704357 and 704393, consecutive primes with equal distance of d = 36.

Links

Crossrefs

Subsequence of A134117.
Cf. A001223, A033451, A047948, A052195.

Programs

  • Mathematica
    Select[Partition[Prime[Range[255000]],3,1],Differences[#]=={36,36}&][[All,1]] (* Harvey P. Dale, Feb 16 2018 *)
  • PARI
    is(n)=nextprime(n+1)==n+36 && nextprime(n+37)==n+72 && isprime(n) \\ Charles R Greathouse IV, Jan 07 2013

Extensions

New name from Charles R Greathouse IV, Jan 07 2013

A052198 Primes p such that p, p+42, p+84 are consecutive primes.

Original entry on oeis.org

247099, 689467, 1008617, 1629767, 1658627, 2024647, 2750999, 2811719, 2880907, 2921777, 3264449, 3295027, 3311317, 3365449, 3555269, 3668419, 4059229, 4412099, 4440529, 4549309, 4619357, 4690219, 4802947, 4955179, 5115259
Offset: 1

Views

Author

Labos Elemer, Jan 28 2000

Keywords

Comments

Old name was: Primes p(k) such that p(k+2)-p(k+1)=p(k+1)-p(k)=42.

Links

Crossrefs

Cf. A001223, A033451, A047948, A052197.

Programs

  • Mathematica
    Select[Partition[Prime[Range[400000]],3,1],Differences[#]=={42,42}&][[All,1]] (* Harvey P. Dale, May 28 2017 *)
  • PARI
    is_A052198(n)=nextprime(n+1)==n+42 && nextprime(n+43)==n+84 && isprime(n) \\ Charles R Greathouse IV, Jan 07 2013, typo corrected by M. F. Hasler, Jan 13 2013

Extensions

New name from Charles R Greathouse IV, Jan 07 2013

A052187 a(n) is the smallest prime p such that p, p+d, and p+2d are consecutive primes where d = 2 for n = 1 and d = 6*(n-1) for n > 1.

Original entry on oeis.org

3, 47, 199, 20183, 16763, 69593, 255767, 247099, 3565931, 6314393, 4911251, 12012677, 23346737, 43607351, 34346203, 36598517, 51041957, 460475467, 652576321, 742585183, 530324329, 807620651, 2988119207, 12447231761, 383204539, 4470607951, 5007182707
Offset: 1

Views

Author

Labos Elemer, Jan 28 2000

Keywords

Comments

The first term 3 is anomalous since for all others d is divisible by 6. These are minimal terms if in A047948 d=6 is replaced by possible differences: (2), 6, 12, 18, ..., 54, 60.
a(54) > 5*10^13, while a(55) = 46186474937633. - Giovanni Resta, Apr 08 2013

Examples

			a(2)=47 and it is the lower border of a dd pattern: 47[6 ]53[6 ]59. a(10)=6314393 and a(10)+54=6314447, a(10)+108=6314501 are consecutive primes and 6314393 is the smallest prime prior to a (54,54) difference pattern of A001223.

Links

Crossrefs

Cf. A001223, A047948, A052160, A054342.
Cf. A052188, A052189, A052195, A052196, A052197, A052198.

Programs

  • Mathematica
    a = Table[0, {100}]; NextPrime[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; p = q = r = 0; Do[r = NextPrime[r]; If[r + p == 2q && r - q < 201 && a[[(r - q)/2]] == 0, a[[(r - q)/2]] = p]; p = q; q = r, {n, 1, 10^6}]; a (* Typos fixed by Zak Seidov, May 01 2020 *)
  • PARI
    list(n)=ve=vector(n);ppp=2;pp=3;forprime(p=5,,d=p-pp;if(pp-ppp==d,i=d\6+1;if(i<=n&&ve[i]==0,ve[i]=ppp;print1(".");vecprod(ve)>0&&return(ve)));ppp=pp;pp=p) \\ Jeppe Stig Nielsen, Apr 17 2022

Formula

The least prime(k) such that prime(k+1) = (prime(k) + prime(k+2))/2 and prime(k+1) - prime(k) = d is either 2 or divisible by 6.
a(1) = A054342(1) - 2. For n>1, a(n) = A054342(n) - 6*(n-1). - Jeppe Stig Nielsen, Apr 16 2022

Extensions

More terms from Labos Elemer, Jan 04 2002
More terms from Robert G. Wilson v, Jan 06 2002
Definition clarified by Harvey P. Dale, Aug 29 2012
a(23)-a(27) from Donovan Johnson, Aug 30 2012
Name edited by Jon E. Schoenfield, Nov 30 2023

A053070 Primes p such that p-6, p and p+6 are consecutive primes.

Original entry on oeis.org

53, 157, 173, 257, 263, 373, 563, 593, 607, 653, 733, 947, 977, 1103, 1123, 1187, 1223, 1367, 1747, 1753, 1907, 2287, 2417, 2677, 2903, 2963, 3307, 3313, 3637, 3733, 4013, 4457, 4597, 4657, 4993, 5107, 5113, 5303, 5387, 5393, 5563, 5807, 6073, 6263
Offset: 1

Views

Author

Harvey P. Dale, Feb 25 2000

Keywords

Comments

Balanced primes separated from the next lower and next higher prime neighbors by 6.
Subset of A006489. - R. J. Mathar, Apr 11 2008
Subset of A006562. - Zak Seidov, Feb 14 2013
a(n) == {3,7} mod 10. - Zak Seidov, Feb 14 2013
Minimal difference is 6: a(5) - a(4) = 263 - 257, a(20) - a(19) = 1753 - 1747, ... . - Zak Seidov, Feb 14 2013

Examples

			157 is separated from both the next lower prime, 151 and the next higher prime, 163, by 6.

Links

Crossrefs

Cf. A047948, A006489, A006562. - Zak Seidov, Feb 14 2013

Programs

  • Maple
    for i from 1 by 1 to 800 do if ithprime(i+1) = ithprime(i) + 6 and ithprime(i+2) = ithprime(i) + 12 then print(ithprime(i+1)); fi; od; # Zerinvary Lajos, Apr 27 2007
  • Mathematica
    lst={};Do[p=Prime[n];If[p-Prime[n-1]==Prime[n+1]-p==6,AppendTo[lst,p]],{n,2,7!}];lst (* Vladimir Joseph Stephan Orlovsky, May 20 2010 *)
Transpose[Select[Partition[Prime[Range[1000]],3,1],Differences[#]=={6,6}&]][[2]] (* Harvey P. Dale, Oct 11 2012 *)

Formula

a(n) = A047948(n) + 6. - R. J. Mathar, Apr 11 2008

Extensions

Edited by N. J. A. Sloane at the suggestion of Zak Seidov, Apr 09 2008

A079016 Suppose p and q = p+12 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 14 possible difference patterns, namely [12], [2,10], [4,8], [6,6], [8,4], [10,2], [2,4,6], [2,6,4], [4,2,6], [4,6,2], [6,2,4], [6,4,2], [2,4,2,4] and [4,2,4,2]. Sequence gives smallest value of p for each difference pattern, sorted by magnitude.

Original entry on oeis.org

5, 7, 17, 19, 29, 31, 47, 67, 89, 137, 139, 199, 397, 1601
Offset: 1

Views

Author

Labos Elemer, Jan 24 2003

Keywords

Examples

			p=1601, q=1613 has difference pattern [6,2,4] and {1601,1607,1609,1613} is the corresponding consecutive prime 4-tuple.

Crossrefs

A022006(1)=5, A022007(1)=7, A078847(1)=17, A078851(1)=19, A078848(1)=29, A078855(1)=31, A047948(1)=47, A078850(1)=67, A031930(1)=A000230(6)=199, A046137(1)=7, A078853(1)=1601.
Cf. A079017-A079024.

Programs

  • Mathematica
    Function[s, Function[t, Union@ Flatten@ Map[s[[First@ Position[t, #]]] &, {{12}, {2, 10}, {4, 8}, {6, 6}, {8, 4}, {10, 2}, {2, 4, 6}, {2, 6, 4}, {4, 2, 6}, {4, 6, 2}, {6, 2, 4}, {6, 4, 2}, {2, 4, 2, 4}, {4, 2, 4, 2}}]]@ Map[Differences@ Select[Range[#, # + 12], PrimeQ] &, s]]@ Select[Prime@ Range[10^3], PrimeQ[# + 12] &] (* Michael De Vlieger, Feb 25 2017 *)

A078561 p, p+4 and p+10 are consecutive primes.

Original entry on oeis.org

19, 43, 79, 127, 163, 229, 349, 379, 439, 499, 643, 673, 937, 967, 1009, 1093, 1213, 1279, 1429, 1489, 1549, 1597, 1609, 2203, 2347, 2389, 2437, 2539, 2689, 2833, 2953, 3079, 3319, 3529, 3613, 3793, 3907, 3919, 4003, 4129, 4447, 4639, 4789, 4933, 4999
Offset: 1

Views

Author

Labos Elemer, Dec 10 2002

Keywords

Comments

Subsequence of A029710. - R. J. Mathar, May 06 2017

Examples

			Between p and p+10 [46] difference-pattern: 19(4)23(6)29;

Links

Crossrefs

Cf. analogous inter-prime d-patterns with d<=6: A022004[24], A022005[42], A049437[26], A049438[62], A078561[46], A078562[64], A047948[66].

Programs

  • Mathematica
    Select[Prime@ Range[10^3], Differences@ NestList[NextPrime, #, 2] == {4, 6} &] (* Michael De Vlieger, May 06 2017 *)
Select[Partition[Prime[Range[700]],3,1],Differences[#]=={4,6}&][[All,1]] (* Harvey P. Dale, Mar 24 2018 *)
  • PARI
    isok(p) = isprime(p) && (nextprime(p+1) == p+4) && (nextprime(p+5) == p+10); \\ Michel Marcus, Dec 20 2013
  • PARI
    is(n)=isprime(n) && isprime(n+4) && isprime(n+10) && !isprime(n+6) && n>3 \\ Charles R Greathouse IV, Dec 20 2013

A078562 p, p+6 and p+10 are consecutive primes.

Original entry on oeis.org

31, 61, 73, 157, 271, 373, 433, 607, 733, 751, 1291, 1543, 1657, 1777, 1861, 1987, 2131, 2287, 2341, 2371, 2383, 2467, 2677, 2791, 2851, 3181, 3313, 3607, 3691, 4441, 4507, 4723, 4993, 5407, 5431, 5521, 5563, 5641, 5683, 5851, 6037, 6211, 6571, 6961
Offset: 1

Views

Author

Labos Elemer, Dec 10 2002

Keywords

Comments

Subsequence of A031924. - R. J. Mathar, Jun 15 2013

Examples

			Between p and p+10 the difference-pattern is [64] like e.g. for p=31: 31(6)37(4)41.

Links

Crossrefs

Cf. analogous inter-prime d-patterns with d<=6: A022004[24], A022005[42], A049437[26], A049438[62], A078561[46], A078562[64], A047948[66].

Programs

  • Mathematica
    Transpose[Select[Partition[Prime[Range[1000]],3,1],#[[3]]-#[[1]]==10&&#[[2]]-#[[1]]==6&]][[1]] (* Harvey P. Dale, Dec 09 2010 *)

A052377 Primes followed by an [8,4,8]=[d,D-d,d] prime difference pattern of A001223.

Original entry on oeis.org

389, 479, 1559, 3209, 8669, 12269, 12401, 13151, 14411, 14759, 21851, 28859, 31469, 33191, 36551, 39659, 40751, 50321, 54311, 64601, 70229, 77339, 79601, 87671, 99551, 102539, 110261, 114749, 114761, 118661, 129449, 132611, 136511
Offset: 1

Views

Author

Labos Elemer, Mar 22 2000

Keywords

Comments

A subsequence of A031926. [Corrected by Sean A. Irvine, Nov 07 2021]
a(n)=p, the initial prime of two consecutive 8-twins of primes as follows: [p,p+8] and [p+12,p+12+8], d=8, while the distance of the two 8-twins is 12 (minimal; see A052380(4/2)=12).
Analogous sequences are A047948 for d=2, A052378 for d=4, A052376 for d=10 and A052188-A052199 for d=6k, so that in the [d,D-d,d] difference patterns which follows a(n) the D-d is minimal(=0,2,4; here it is 4).

Examples

			p=1559 begins the [1559,1567,1571,1579] prime quadruple consisting of two 8-twins [1559,1567] and[1571,1579] which are in minimal distance, min{D}=1571-1559=12=A052380(8/2).

Crossrefs

Cf. A031926, A053325, A052380, A052376, A052378, A052188-A052190.

Formula

a(n) is the initial term of a [p, p+8, p+12, p+12+8] quadruple of consecutive primes.
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