A022005 -id:A022005 - cp's OEIS frontend

A052378 Primes followed by a [4,2,4] prime difference pattern of A001223.

7, 13, 37, 97, 103, 223, 307, 457, 853, 877, 1087, 1297, 1423, 1483, 1867, 1993, 2683, 3457, 4513, 4783, 5227, 5647, 6823, 7873, 8287, 10453, 13687, 13873, 15727, 16057, 16063, 16183, 17383, 19417, 19423, 20743, 21013, 21313, 22273, 23053, 23557

Offset: 1

Labos Elemer, Mar 22 2000

nonn

The sequence includes A052166, A052168, A022008 and also other primes like 13, 103, 16063 etc.
a(n) is the lesser term of a 4-twin (A023200) after which the next 4-twin comes in minimal distance [here it is 2; see A052380(4/2)].
Analogous prime sequences are A047948, A052376, A052377 and A052188-A052198 with various [d, A052380(d/2), d] difference patterns following a(n).
All terms == 1 (mod 6) - Zak Seidov, Aug 27 2012
Subsequence of A022005. - R. J. Mathar, May 06 2017

			103 initiates [103,107,109,113] prime quadruple followed by [4,2,4] difference pattern.

Zak Seidov, Table of n, a(n) for n = 1..2000

Cf. A023200, A053320, A022008, A052166, A052168, A001223, A052380.

a = {}; Do[If[Prime[x + 3] - Prime[x] == 10, AppendTo[a, Prime[x]]], {x, 1, 10000}]; a (* Zerinvary Lajos, Apr 03 2007 *)
Select[Partition[Prime[Range[3000]],4,1],Differences[#]=={4,2,4}&][[All,1]] (* Harvey P. Dale, Jun 16 2017 *)

is(n)=n%6==1 && isprime(n+4) && isprime(n+6) && isprime(n+10) && isprime(n) \\ Charles R Greathouse IV, Apr 29 2015

a(n) is the initial prime of a [p, p+4, p+6, p+6+4] prime-quadruple consisting of two 4-twins: [p, p+4] and [p+6, p+10].

A039669 Numbers n > 2 such that n - 2^k is a prime for all k > 0 with 2^k < n.

Original entry on oeis.org

4, 7, 15, 21, 45, 75, 105

Offset: 1

Felice Russo

Erdős conjectures that these are the only values of n with this property.
No other terms below 2^120. - Max Alekseyev, Dec 08 2011
Curiously, Mientka and Weitzenkamp say there are 9 such numbers below 20000. - Michel Marcus, May 12 2013
Presumably, Mientka and Weitzenkamp are including 1 and 2. - Robert Israel, Dec 23 2015
Observation: The prime numbers of the form (n-2) associated with each element of the series are (2,5,13,19,43,73,103). These prime numbers are exactly the first elements of A068374 (primes n such that positive values of n - A002110(k) are all primes for k>0). - David Morales Marciel, Dec 14 2015

			45 is here because 43, 41, 37, 29 and 13 are primes.

R. K. Guy, Unsolved Problems in Number Theory, A19.
F. Le Lionnais, Les Nombres Remarquables, Paris, Hermann, 1983, p. 96, 1983.
D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, p. 306.
D. Wells, Curious and interesting numbers, Penguin Books, p. 118.

P. Erdős, On integers of the form 2^k + p and some related questions, Summa Bras. Math., 2 (1950), 113-123.
Walter E. Mientka and Roger C. Weitzenkamp, On f-plentiful numbers, Journal of Combinatorial Theory, Volume 7, Issue 4, December 1969, pages 374-377.

Cf. A067526 (n such that n-2^k is prime or 1), A067527 (n such that n-3^k is prime), A067528 (n such that n-4^k is prime or 1), A067529 (n such that n-5^k is prime), A100348 (n such that n-4^k is prime), A100349 (n such that n-2^k is prime or semiprime), A100350 (primes p such that p-2^k is prime or semiprime), A100351 (n such that n-2^k is semiprime).

Cf. A022005.

N = 10^8; % to get terms < N
p = primes(N);
A = [3:N];
for k = 1:floor(log2(N))
  A = intersect(A, [1:(2^k), (p+2^k)]);
end
A % Robert Israel, Dec 23 2015

lst={}; Do[k=1; While[p=n-2^k; p>0 && PrimeQ[p], k++ ]; If[p<=0, AppendTo[lst, n]], {n, 3, 1000}]; lst (* T. D. Noe, Sep 15 2002 *)

isok(n) = {my(k = 1); while (2^k < n, if (! isprime(n-2^k), return (0)); k++;); return (1);} \\ Michel Marcus, Dec 14 2015

Additional comments from T. D. Noe, Sep 15 2002

Definition edited by Robert Israel, Dec 23 2015

A078850 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=[4,2,6]; short d-string notation of pattern = [426].

Original entry on oeis.org

67, 1447, 2377, 2707, 5437, 5737, 7207, 9337, 11827, 12037, 19207, 21487, 21517, 23197, 26107, 26947, 28657, 31147, 31177, 35797, 37357, 37567, 42697, 50587, 52177, 65167, 67927, 69997, 71707, 74197, 79147, 81547, 103087, 103387, 106657

Offset: 1

Labos Elemer, Dec 11 2002

nonn

Subsequence of A022005. - R. J. Mathar, May 06 2017

			p=67,67+4=71,67+4+2=73,67+4+2+6=79 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 = {4, 2, 6}; First /@ Select[Partition[Prime@ Range@ 12000, Length@ d + 1, 1], Differences@ # == d &] (* Michael De Vlieger, May 02 2016 *)

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

Listed terms verified by Ray Chandler, Apr 20 2009

A201596 Record (maximal) gaps between prime triples (p, p+4, p+6).

Original entry on oeis.org

6, 24, 30, 90, 150, 156, 210, 240, 306, 366, 384, 444, 810, 834, 1086, 1200, 1326, 2316, 3876, 4230, 4350, 8244, 8880, 9450, 10686, 10950, 11784, 12816, 13554, 15504, 15576, 16254, 16506, 16596, 19446, 19944, 21516, 38340, 39990, 41556, 45786, 47190, 48246, 59856

Offset: 1

Alexei Kourbatov, Dec 03 2011

nonn

Prime triples (p, p+4, p+6) are one of the two types of densest permissible constellations of 3 primes (A022004 and A022005). By the Hardy-Littlewood k-tuple conjecture, average gaps between prime k-tuples are O(log^k(p)), with k=3 for triples. If a gap is larger than any preceding gap, we call it a maximal gap, or a record gap. Maximal gaps may be significantly larger than average gaps; this sequence suggests that maximal gaps between triples are O(log^4(p)).

A201597 lists initial primes p in triples (p, p+4, p+6) preceding the maximal gaps. A233435 lists the corresponding primes p at the end of the maximal gaps.

			The gap of 6 between triples starting at p=7 and p=13 is the very first gap, so a(1)=6. The gap of 24 between triples starting at p=13 and p=37 is a maximal gap - larger than any preceding gap; therefore a(2)=24. The gap of 30 between triples at p=37 and p=67 is again a maximal gap, so a(3)=30. The next gap is smaller, so it does not contribute to the sequence.

Alexei Kourbatov, Table of n, a(n) for n = 1..79
G. H. Hardy and J. E. Littlewood, Some problems of 'Partitio numerorum'; III: on the expression of a number as a sum of primes, Acta Mathematica, Vol. 44, pp. 1-70, 1923.
Alexei Kourbatov, Maximal gaps between prime k-tuples
A. Kourbatov, Maximal gaps between prime k-tuples: a statistical approach, arXiv preprint arXiv:1301.2242 [math.NT], 2013 and J. Int. Seq. 16 (2013) #13.5.2
Alexei Kourbatov, Tables of record gaps between prime constellations, arXiv preprint arXiv:1309.4053 [math.NT], 2013.
Alexei Kourbatov, The distribution of maximal prime gaps in Cramer's probabilistic model of primes, arXiv preprint arXiv:1401.6959 [math.NT], 2014.
Alexei Kourbatov and Marek Wolf, Predicting maximal gaps in sets of primes, arXiv preprint arXiv:1901.03785 [math.NT], 2019.
Norman Luhn, Record Gaps Between Prime Triplets.
Eric W. Weisstein, k-Tuple Conjecture

Cf. A022005 (prime triples p, p+4, p+6), A113274, A113404, A200503, A201598, A201062, A201073, A201051, A201251, A202281, A202361, A201597, A233435.

DeleteDuplicates[Differences[Select[Partition[Prime[Range[5*10^6]],3,1],Differences[#]=={4,2}&][[;;,1]]],GreaterEqual]  (* Harvey P. Dale, Feb 26 2023 *)

Gaps between prime triples (p, p+4, p+6) are smaller than 0.35*(log p)^4, where p is the prime at the end of the gap. There is no rigorous proof of this formula. The O(log^4(p)) growth rate is suggested by numerical data and heuristics based on probability considerations.

A201598 Record (maximal) gaps between prime triples (p, p+2, p+6).

Original entry on oeis.org

6, 24, 60, 84, 114, 180, 210, 264, 390, 564, 630, 1050, 1200, 1530, 2016, 2844, 3426, 3756, 3864, 3936, 4074, 4110, 6090, 8250, 9240, 9270, 10344, 10506, 10734, 10920, 12930, 15204, 20190, 20286, 21216, 25746, 34920, 38820, 39390, 41754, 43020, 44310, 52500, 71346

Offset: 1

Alexei Kourbatov, Dec 03 2011

nonn

Prime triples (p, p+2, p+6) are one of the two types of densest permissible constellations of 3 primes (A022004 and A022005). By the Hardy-Littlewood k-tuple conjecture, average gaps between prime k-tuples are O(log^k(p)), with k=3 for triples. If a gap is larger than any preceding gap, we call it a maximal gap, or a record gap. Maximal gaps may be significantly larger than average gaps; this sequence suggests that maximal gaps between triples are O(log^4(p)).

A201599 lists initial primes p in triples (p, p+2, p+6) preceding the maximal gaps. A233434 lists the corresponding primes p at the end of the maximal gaps.

			The gap of 6 between triples starting at p=5 and p=11 is the very first gap, so a(1)=6. The gap of 6 between triples starting at p=11 and p=17 is not a record, so it does not contribute to the sequence. The gap of 24 between triples starting at p=17 and p=41 is a maximal gap - larger than any preceding gap; therefore a(2)=24.

Alexei Kourbatov, Table of n, a(n) for n = 1..72
G. H. Hardy and J. E. Littlewood, Some problems of 'Partitio numerorum'; III: on the expression of a number as a sum of primes, Acta Mathematica, Vol. 44, pp. 1-70, 1923.
Alexei Kourbatov, Maximal gaps between prime k-tuples
A. Kourbatov, Maximal gaps between prime k-tuples: a statistical approach, arXiv preprint arXiv:1301.2242 [math.NT], 2013 and J. Int. Seq. 16 (2013) #13.5.2
Alexei Kourbatov, Tables of record gaps between prime constellations, arXiv preprint arXiv:1309.4053 [math.NT], 2013.
Alexei Kourbatov, The distribution of maximal prime gaps in Cramer's probabilistic model of primes, arXiv preprint arXiv:1401.6959 [math.NT], 2014.
Alexei Kourbatov and Marek Wolf, Predicting maximal gaps in sets of primes, arXiv preprint arXiv:1901.03785 [math.NT], 2019.
Norman Luhn, Record Gaps Between Prime Triplets.
Eric Weisstein's World of Mathematics, k-Tuple Conjecture.

Cf. A022004 (prime triples p, p+2, p+6), A113274, A113404, A200503, A201596, A201062, A201073, A201051, A201251, A202281, A202361, A201599, A233434.

Gaps between prime triples (p, p+2, p+6) are smaller than 0.35*(log p)^4, where p is the prime at the end of the gap. There is no rigorous proof of this formula. The O(log^4(p)) growth rate is suggested by numerical data and heuristics based on probability considerations.

A098413 Greatest members p of prime triples (p-6, p-2, p).

Original entry on oeis.org

13, 19, 43, 73, 103, 109, 199, 229, 283, 313, 463, 619, 829, 859, 883, 1093, 1303, 1429, 1453, 1489, 1669, 1699, 1789, 1873, 1879, 1999, 2089, 2143, 2383, 2689, 2713, 2803, 3169, 3259, 3463, 3469, 3853, 4159, 4519, 4789, 5233, 5419, 5443, 5653, 5659, 5743

Offset: 1

Reinhard Zumkeller, Sep 07 2004

nonn

Subsequence of A046117; a(n) = A073649(n) + 2 = A022005(n) + 6.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Prime Triplet

Cf. A098412, A098415.

[p: p in PrimesUpTo(6500)|IsPrime(p) and IsPrime(p-6) and IsPrime(p-2)]; // Vincenzo Librandi, Dec 26 2010

Transpose[Select[Partition[Prime[Range[800]],3,1],Differences[#] == {4,2}&]][[3]] (* Harvey P. Dale, Aug 21 2013 *)

A237769 Number of primes p < n with pi(n-p) - 1 and pi(n-p) + 1 both prime, where pi(.) is given by A000720.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 2, 3, 2, 2, 3, 3, 3, 4, 3, 4, 4, 2, 2, 2, 2, 4, 4, 2, 2, 2, 2, 3, 3, 1, 1, 2, 2, 3, 4, 3, 3, 4, 3, 5, 5, 3, 3, 2, 2, 5, 5, 3, 3, 3, 3, 5, 5, 2, 2, 3, 3, 3, 4, 2, 2, 6, 6, 9, 8, 4, 4, 3, 3, 6, 6, 5, 5, 4, 4, 7

Offset: 1

Zhi-Wei Sun, Feb 13 2014

nonn

Conjecture: (i) a(n) > 0 for all n > 8, and a(n) = 1 only for n = 9, 34, 35.
(ii) For any integer n > 4, there is a prime p < n such that 3*pi(n-p) - 1, 3*pi(n-p) + 1 and 3*pi(n-p) + 5 are all prime. Also, for each integer n > 8, there is a prime p < n such that 3*pi(n-p) - 1, 3*pi(n-p) + 1 and 3*pi(n-p) - 5 are all prime.
(iii) For any integer n > 6, there is a prime p < n such that phi(n-p) - 1 and phi(n-p) + 1 are both prime, where phi(.) is Euler's totient function.

			a(9) = 1 since 2, pi(9-2) - 1 = 3 and pi(9-2) + 1 = 5 are all prime.
a(34) = 1 since 19, pi(34-19) - 1 = pi(15) - 1 = 5 and pi(34-19) + 1 = pi(15) + 1 = 7 are all prime.
a(35) = 1 since 19, pi(35-19) - 1 = pi(16) - 1 = 5 and pi(35-19) + 1 = pi(16) + 1 = 7 are all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000010, A000040, A000720, A001359, A006512, A014574, A022004, A022005, A237705, A237706, A237768.

TQ[n_]:=PrimeQ[n-1]&&PrimeQ[n+1]
a[n_]:=Sum[If[TQ[PrimePi[n-Prime[k]]],1,0],{k,1,PrimePi[n-1]}]
Table[a[n],{n,1,80}]

A073649 Define the composite field of a prime q to be f(q) = (t,s) if p, q, r are three consecutive primes and q-p = t and r-q = s. This is a sequence of primes q with field (4,2).

Original entry on oeis.org

11, 17, 41, 71, 101, 107, 197, 227, 281, 311, 461, 617, 827, 857, 881, 1091, 1301, 1427, 1451, 1487, 1667, 1697, 1787, 1871, 1877, 1997, 2087, 2141, 2381, 2687, 2711, 2801, 3167, 3257, 3461, 3467, 3851, 4157, 4517, 4787, 5231, 5417, 5441, 5651, 5657

Offset: 1

Amarnath Murthy, Aug 09 2002

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A073648, A073650.
Cf. A098413, A098414.
Equals A022005 + 4.

Transpose[Select[Partition[Prime[Range[1200]],3,1],Differences[#] == {4,2}&]] [[2]] (* Harvey P. Dale, Jul 23 2011 *)

Corrected and extended by Benoit Cloitre, Aug 13 2002

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 *)

cp's OEIS Frontend

A052378 Primes followed by a [4,2,4] prime difference pattern of A001223.

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A039669 Numbers n > 2 such that n - 2^k is a prime for all k > 0 with 2^k < n.

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A078850 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=[4,2,6]; short d-string notation of pattern = [426].

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A201596 Record (maximal) gaps between prime triples (p, p+4, p+6).

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A201598 Record (maximal) gaps between prime triples (p, p+2, p+6).

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A098413 Greatest members p of prime triples (p-6, p-2, p).

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A237769 Number of primes p < n with pi(n-p) - 1 and pi(n-p) + 1 both prime, where pi(.) is given by A000720.

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A073649 Define the composite field of a prime q to be f(q) = (t,s) if p, q, r are three consecutive primes and q-p = t and r-q = s. This is a sequence of primes q with field (4,2).