A049437 -id:A049437 - cp's OEIS frontend

A023202 Primes p such that p + 8 is also prime.

3, 5, 11, 23, 29, 53, 59, 71, 89, 101, 131, 149, 173, 191, 233, 263, 269, 359, 389, 401, 431, 449, 479, 491, 563, 569, 593, 599, 653, 683, 701, 719, 743, 761, 821, 911, 929, 983, 1013, 1031, 1061, 1109, 1163, 1193, 1223, 1229, 1283, 1289, 1319, 1373, 1439

Offset: 1

David W. Wilson

All terms > 3 are congruent to 5 mod 6 (observation by Zak Seidov in SeqFan). Thus each corresponding p + 8 is congruent to 1 mod 6. - Rick L. Shepherd, Mar 25 2023

Matt C. Anderson, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe, corrected by Sean A. Irvine and Georg Fischer)
A. Granville and G. Martin, Prime number races, arXiv:math/0408319 [math.NT], 2004.
Maxie D. Schmidt, New Congruences and Finite Difference Equations for Generalized Factorial Functions, arXiv:1701.04741 [math.CO], 2017.
Eric Weisstein's World of Mathematics, Twin Primes

Disjoint union of A007530, A031926, A049437, A049438.

Cf. A046134, A049436, A046138, A015915.

Filtered([1..1500], k-> IsPrime(k) and IsPrime(k+8)); # G. C. Greubel, Feb 07 2020

[n: n in [0..1500] | IsPrime(n) and IsPrime(n+8)]; // Vincenzo Librandi, Nov 20 2010

select(n-> isprime(n) and isprime(n+8), [`$`(1..1500)]); # G. C. Greubel, Feb 07 2020

Select[Range[1500], PrimeQ[#] && PrimeQ[#+8]&] (* Vladimir Joseph Stephan Orlovsky, Aug 29 2008 *)
Select[Prime[Range[250]],PrimeQ[#+8]&] (* Harvey P. Dale, Dec 24 2020 *)

is(n)=isprime(n)&&isprime(n+8) \\ Charles R Greathouse IV, Jul 01 2013

[n for n in (1..1500) if is_prime(n) and is_prime(n+8)] # G. C. Greubel, Feb 07 2020

A078848 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=[2,6,4]; short d-string notation of pattern = [264].

Original entry on oeis.org

29, 59, 71, 269, 431, 1289, 2129, 2339, 2381, 2789, 4721, 5519, 5639, 5849, 6569, 6959, 8999, 10091, 13679, 14549, 16649, 16691, 18119, 19379, 19751, 21491, 25931, 27689, 27791, 28619, 31181, 32369, 32561, 32831, 36779, 41609, 43961, 45119

Offset: 1

Labos Elemer, Dec 11 2002

nonn

Subsequence of A049437. - R. J. Mathar, Feb 10 2013

			29, 29+2=31, 29+2+6=37, 29+2+6+4=41 are consecutive primes.

R. J. Mathar, Table of n, a(n) for n = 1..1000

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].

d = {2, 6, 4}; First /@ Select[Partition[Prime@ Range[10^4], Length@ d + 1, 1], Differences@ # == d &] (* Michael De Vlieger, May 02 2016 *)
Select[Partition[Prime[Range[4700]],4,1],Differences[#]=={2,6,4}&][[All,1]] (* Harvey P. Dale, Mar 08 2020 *)

Primes p=p(i) such that p(i+1)=p+2, p(i+2)=p+2+6, p(i+3)=p+2+6+4.

Listed terms verified by Ray Chandler, Apr 20 2009

Typo in example corrected by Michel Marcus, Dec 28 2013

A078849 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=[2, 6,6]; short d-string notation of pattern = [266].

Original entry on oeis.org

149, 599, 3299, 4649, 5099, 6359, 11489, 12539, 16979, 19469, 27059, 30089, 31319, 34259, 42179, 53609, 58229, 63689, 65699, 71339, 75209, 77549, 78569, 80909, 81929, 85829, 87509, 87539, 89519, 92219, 101279, 105359, 112289, 116099, 116789

Offset: 1

Labos Elemer, Dec 11 2002

nonn

Subsequence of A049437. - R. J. Mathar, Feb 10 2013

			149, 149+2=151, 149+2+6=157, 149+2+6+6=163 are consecutive primes.

R. J. Mathar, Table of n, a(n) for n = 1..1000

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].

d = {2, 6, 6}; First /@ Select[Partition[Prime@ Range@ 12000, Length@ d + 1, 1], Differences@ # == d &] (* Michael De Vlieger, May 02 2016 *)
Select[Partition[Prime[Range[12000]],4,1],Differences[#]=={2,6,6}&][[All,1]] (* Harvey P. Dale, Dec 29 2017 *)

Primes p = p(i) such that p(i+1)=p+2, p(i+2)=p+2+6, p(i+3)=p+2+6+6.

Listed terms verified by Ray Chandler, Apr 20 2009

A049438 p, p+6 and p+8 are all primes (A046138) but p+2 is not.

Original entry on oeis.org

23, 53, 131, 173, 233, 263, 563, 593, 653, 1013, 1223, 1283, 1601, 1613, 2333, 2543, 2963, 3323, 3533, 3761, 3911, 3923, 4013, 4211, 4253, 4643, 4793, 5003, 5273, 5471, 5843, 5861, 6263, 6353, 6563, 6653, 6863, 7121, 7451, 7481, 7541, 7583

Offset: 1

Labos Elemer

R. J. Mathar, Table of n, a(n) for n = 1..1000

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

Cf. A007530, A023202, A031926, A046134, A046138, A049436, A049437.

Select[Prime@ Range[10^3], MatchQ[Boole@ PrimeQ@ {# + 2, # + 6, # + 8}, {0, 1, 1}] &] (* Michael De Vlieger, Feb 05 2017 *)

isok(p) = isprime(p) && !isprime(p+2) && isprime(p+6) && isprime(p+8); \\ Michel Marcus, Dec 13 2013

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

Labos Elemer, Dec 10 2002

nonn

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

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

R. J. Mathar, Table of n, a(n) for n = 1..1000

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

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 *)

isok(p) = isprime(p) && (nextprime(p+1) == p+4) && (nextprime(p+5) == p+10); \\ Michel Marcus, Dec 20 2013

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

Labos Elemer, Dec 10 2002

nonn

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

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

R. J. Mathar, Table of n, a(n) for n = 1..1000

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

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

A049436 p, p+8 and either p+2 or p+6 or both are all primes.

Original entry on oeis.org

3, 5, 11, 23, 29, 53, 59, 71, 101, 131, 149, 173, 191, 233, 263, 269, 431, 563, 569, 593, 599, 653, 821, 1013, 1031, 1061, 1223, 1229, 1283, 1289, 1319, 1451, 1481, 1601, 1613, 1619, 1871, 2081, 2129, 2333, 2339, 2381, 2543, 2549, 2711, 2789, 2963, 3251

Offset: 1

Labos Elemer

nonn

			3 is here because 5, 7 and 11 are primes; 5 is here because 7, 11 and 13 are primes.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

A031926, A023202, A049437, A049438, A046134, A046138, A007530.

Select[Prime[Range[500]],PrimeQ[#+8]&&AnyTrue[#+{2,6},PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jan 04 2017 *)

A181994 Initial members of prime triples p < q < r such that r-q = n*(q-p).

Original entry on oeis.org

3, 2, 29, 8117, 137, 197, 45433, 1931, 521, 156151, 1949, 1667, 480203, 2969, 7757, 2181731, 12161, 28349, 6012893, 20807, 16139, 3933593, 163061, 86627, 13626251, 25469, 40637, 60487753, 79697, 149627, 217795241, 625697, 552401, 240485251, 173357, 360089, 122164741

Offset: 1

Zak Seidov, May 31 2012

nonn

For some n, a(n) are abnormally large: note, e.g., that if q-p=2, then n cannot be of the form 4+3k, that is why a(4), a(7), a(10), ... are larger than neighbor terms; also, a(67) > 1.1*10^11. Is the sequence infinite?

			First 10 cases of {n,p,q,r}: {1,3,5,7}, {2,2,3,5}, {3,29,31,37}, {4,8117,8123,8147}, {5,137,139,149}, {6,197,199,211}, {7,45433,45439,45481}, {8,1931,1933,1949}, {9,521,523,541}, {10,156151,156157,156217}.

Zak Seidov, Table of n, a(n) for n = 1..66

Particular cases with q-p=2: A022004 [(r-q)=2*(q-p)], A049437 [(r-q)=3*(q-p) starting with 2nd term].

Cf. A179210.

a(n) = prevprime(A179210(n)). - Robert G. Wilson v, Dec 23 2016

A345213 Primes p such that q^r == r^q (mod p), where p,q,r are consecutive primes.

Original entry on oeis.org

2, 11, 29, 59, 71, 149, 269, 431, 569, 599, 727, 1031, 1061, 1229, 1289, 1319, 1451, 1619, 2129, 2339, 2381, 2549, 2711, 2789, 3299, 3539, 4019, 4049, 4091, 4649, 4721, 5099, 5441, 5519, 5639, 5741, 5849, 6269, 6359, 6569, 6701, 6959, 7211, 8009, 8999, 9041, 9341, 10091, 10859, 11489, 11831

Offset: 1

J. M. Bergot and Robert Israel, Jun 10 2021

nonn

Terms not in A049437 include 2, 11, 727 and 22571. Are there others?

Are there primes p other than 7 such that p^q == q^p (mod r), or primes p other than 41 such that p^r == r^p (mod q), where p,q,r are consecutive primes?

			a(3) = 29 is a term because 29, 31 and 37 are consecutive primes and 37^31 == 31^37 == 19 (mod 29).

Robert Israel, Table of n, a(n) for n = 1..10000

Includes A049437 except for 3.

q:= 2: r:= 3: R:= NULL: count:= 0:
while count < 100 do
  p:= q; q:= r; r:= nextprime(r);
  if q&^r - r&^q mod p = 0 then count:= count+1; R:= R, p fi
od:
R;

A297709 Table read by antidiagonals: Let b be the number of digits in the binary expansion of n. Then T(n,k) is the k-th odd prime p such that the binary digits of n match the primality of the b consecutive odd numbers beginning with p (or 0 if no such k-th prime exists).

Original entry on oeis.org

3, 5, 7, 7, 13, 3, 11, 19, 5, 23, 13, 23, 11, 31, 7, 17, 31, 17, 47, 13, 5, 19, 37, 29, 53, 19, 11, 3, 23, 43, 41, 61, 37, 17, 0, 89, 29, 47, 59, 73, 43, 29, 0, 113, 23, 31, 53, 71, 83, 67, 41, 0, 139, 31, 19, 37, 61, 101, 89, 79, 59, 0, 181, 47, 43, 7, 41, 67

Offset: 1

Jon E. Schoenfield, Apr 15 2018

For each n >= 1, row n is the union of rows 2n and 2n+1.
Rows with no nonzero terms: 15, 21, 23, 28, 30, 31, ...
Rows whose only nonzero term is 3: 7, 14, 29, 59, 118, 237, 475, 950, 1901, 3802, 7604, ...
Rows whose only nonzero term is 5: 219, 438, 877, 1754, 3508, 7017, 14035, ...
For j = 2, 3, 4, ..., respectively, the first row whose only nonzero term is prime(j) is 7, 219, 2921, ...; is there such a row for every odd prime?

			13 = 1101_2, so row n=13 lists the odd primes p such that the four consecutive odd numbers p, p+2, p+4, and p+6 are prime, prime, composite, and prime, respectively; these are the terms of A022004.
14 = 1110_2, so row n=14 lists the odd primes p such that p, p+2, p+4, and p+6 are prime, prime, prime, and composite, respectively; since there is only one such prime (namely, 3), there is no such 2nd, 3rd, 4th, etc. prime, so the terms in row 14 are {3, 0, 0, 0, ...}.
15 = 1111_2, so row n=15 would list the odd primes p such that p, p+2, p+4, and p+6 are all prime, but since no such prime exists, every term in row 15 is 0.
Table begins:
  n in base|                    k                   |  OEIS
  ---------+----------------------------------------+sequence
  10     2 |   1    2    3    4    5    6    7    8 | number
  =========+========================================+========
   1     1 |   3    5    7   11   13   17   19   23 | A065091
   2    10 |   7   13   19   23   31   37   43   47 | A049591
   3    11 |   3    5   11   17   29   41   59   71 | A001359
   4   100 |  23   31   47   53   61   73   83   89 | A124582
   5   101 |   7   13   19   37   43   67   79   97 | A029710
   6   110 |   5   11   17   29   41   59   71  101 | A001359*
   7   111 |   3    0    0    0    0    0    0    0 |
   8  1000 |  89  113  139  181  199  211  241  283 | A083371
   9  1001 |  23   31   47   53   61   73   83  131 | A031924
  10  1010 |  19   43   79  109  127  163  229  313 |
  11  1011 |   7   13   37   67   97  103  193  223 | A022005
  12  1100 |  29   59   71  137  149  179  197  239 | A210360*
  13  1101 |   5   11   17   41  101  107  191  227 | A022004
  14  1110 |   3    0    0    0    0    0    0    0 |
  15  1111 |   0    0    0    0    0    0    0    0 |
  16 10000 | 113  139  181  199  211  241  283  293 | A124584
  17 10001 |  89  359  389  401  449  479  491  683 | A031926
  18 10010 |  31   47   61   73   83  151  157  167 |
  19 10011 |  23   53  131  173  233  263  563  593 | A049438
  20 10100 |  19   43   79  109  127  163  229  313 |
  21 10101 |   0    0    0    0    0    0    0    0 |
  22 10110 |   7   13   37   67   97  103  193  223 | A022005
  23 10111 |   0    0    0    0    0    0    0    0 |
  24 11000 | 137  179  197  239  281  419  521  617 |
  25 11001 |  29   59   71  149  269  431  569  599 | A049437*
  26 11010 |  17   41  107  227  311  347  461  641 |
  27 11011 |   5   11  101  191  821 1481 1871 2081 | A007530
  28 11100 |   0    0    0    0    0    0    0    0 |
  29 11101 |   3    0    0    0    0    0    0    0 |
  30 11110 |   0    0    0    0    0    0    0    0 |
  31 11111 |   0    0    0    0    0    0    0    0 |
*other than the referenced sequence's initial term 3
.
Alternative version of table:
.
  n in base|primal-|               k              |  OEIS
  ---------+  ity  +------------------------------+  seq.
  10     2 |pattern|   1    2    3    4    5    6 | number
  =========+=======+==============================+========
   1     1 | p     |   3    5    7   11   13   17 | A065091
   2    10 | pc    |   7   13   19   23   31   37 | A049591
   3    11 | pp    |   3    5   11   17   29   41 | A001359
   4   100 | pcc   |  23   31   47   53   61   73 | A124582
   5   101 | pcp   |   7   13   19   37   43   67 | A029710
   6   110 | ppc   |   5   11   17   29   41   59 | A001359*
   7   111 | ppp   |   3    0    0    0    0    0 |
   8  1000 | pccc  |  89  113  139  181  199  211 | A083371
   9  1001 | pccp  |  23   31   47   53   61   73 | A031924
  10  1010 | pcpc  |  19   43   79  109  127  163 |
  11  1011 | pcpp  |   7   13   37   67   97  103 | A022005
  12  1100 | ppcc  |  29   59   71  137  149  179 | A210360*
  13  1101 | ppcp  |   5   11   17   41  101  107 | A022004
  14  1110 | pppc  |   3    0    0    0    0    0 |
  15  1111 | pppp  |   0    0    0    0    0    0 |
  16 10000 | pcccc | 113  139  181  199  211  241 | A124584
  17 10001 | pcccp |  89  359  389  401  449  479 | A031926
  18 10010 | pccpc |  31   47   61   73   83  151 |
  19 10011 | pccpp |  23   53  131  173  233  263 | A049438
  20 10100 | pcpcc |  19   43   79  109  127  163 |
  21 10101 | pcpcp |   0    0    0    0    0    0 |
  22 10110 | pcppc |   7   13   37   67   97  103 | A022005
  23 10111 | pcppp |   0    0    0    0    0    0 |
  24 11000 | ppccc | 137  179  197  239  281  419 |
  25 11001 | ppccp |  29   59   71  149  269  431 | A049437*
  26 11010 | ppcpc |  17   41  107  227  311  347 |
  27 11011 | ppcpp |   5   11  101  191  821 1481 | A007530
  28 11100 | pppcc |   0    0    0    0    0    0 |
  29 11101 | pppcp |   3    0    0    0    0    0 |
  30 11110 | ppppc |   0    0    0    0    0    0 |
  31 11111 | ppppp |   0    0    0    0    0    0 |
.
     *other than the referenced sequence's initial term 3

Cf. A001359, A007530, A022004, A022005, A029710, A031924, A031926, A049437, A049438, A049591, A065091, A124582, A083371, A124584, A210360.

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A023202 Primes p such that p + 8 is also prime.

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A078848 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=[2,6,4]; short d-string notation of pattern = [264].

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A078849 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=[2, 6,6]; short d-string notation of pattern = [266].

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A049438 p, p+6 and p+8 are all primes (A046138) but p+2 is not.

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

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

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A049436 p, p+8 and either p+2 or p+6 or both are all primes.

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A181994 Initial members of prime triples p < q < r such that r-q = n*(q-p).