A001223 -id:A001223 - cp's OEIS frontend

A089180 a(n) is the smallest number m such that d(m) = d(m+1) = ... = d(m+n), where d(k) = prime(k+1) - prime(k) (A001223).

2, 54, 654926, 6904737

Offset: 1

Farideh Firoozbakht, Dec 07 2003

a(5) is greater than 105000000.

The a(n)-th prime is the smallest start of n+2 consecutive primes in arithmetic progression. - Jens Kruse Andersen, Jun 14 2014

			a(3) = 659426 because d(659426) = d(659426+1) = d(659426+2) = d(6594286+3) or 9843019, 9843049, 9843079, 9843109, 9843139 are five consecutive primes with same difference and prime(659426) = 9843019 is the smallest prime number with this property.
Also a(4) = 6904737 because d(6904737) = d(6904737+1) = ... = d(6904737+4) or 121174811, 121174841, 121174871, 121174901, 121174931, 121174961 are six consecutive primes with same difference and prime(6904737) = 121174811 is the smallest prime number with this property.

J. K. Andersen, The minimal CPAP-k.
L. J. Lander and T. R. Parkin, Consecutive primes in arithmetic progression, Math. Comp. vol. 21 no. 99 (1967) p. 489.
G. W. Polites, Prime Desert n-Tuplets, Amer. Math. Monthly vol. 95 no. 2 (1988) pp. 98-104.

Cf. A001223, A090403.

A000040[a(n)]=A006560(n+2). - R. J. Mathar, Aug 10 2007

a(n) = A000720(A006560(n+2)). - Jens Kruse Andersen, Jun 14 2014

A333212 Lengths of maximal weakly decreasing subsequences in the sequence of prime gaps (A001223).

Original entry on oeis.org

1, 2, 2, 2, 1, 2, 3, 1, 3, 3, 2, 1, 3, 2, 1, 2, 2, 2, 3, 3, 2, 2, 4, 1, 2, 5, 3, 1, 3, 1, 2, 2, 1, 1, 4, 1, 2, 1, 2, 2, 2, 1, 3, 1, 3, 2, 1, 2, 2, 4, 1, 4, 4, 3, 1, 3, 2, 1, 1, 2, 5, 3, 2, 2, 2, 2, 2, 1, 3, 1, 3, 1, 2, 1, 3, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 1, 3

Offset: 1

Gus Wiseman, Mar 14 2020

nonn

Prime gaps are differences between adjacent prime numbers.

			The prime gaps split into the following weakly decreasing subsequences: (1), (2,2), (4,2), (4,2), (4), (6,2), (6,4,2), (4), (6,6,2), (6,4,2), (6,4), (6), ...

First differences of A258025 (with zero prepended).
The version for the Kolakoski sequence is A332273.
The weakly increasing version is A333215.
The unequal version is A333216.
The strictly decreasing version is A333252.
The strictly increasing version is A333253.
The equal version is A333254.
Prime gaps are A001223.
Positions of adjacent equal differences are A064113.
Weakly decreasing runs of compositions in standard order are A124765.
Positions of strict ascents in the sequence of prime gaps are A258025.
Cf. A000040, A000720, A001221, A036263, A054819, A084758, A114994, A124760, A124761, A124768, A333213, A333214.

Length/@Split[Differences[Array[Prime,100]],#1>=#2&]//Most

Ones correspond to weak prime quartets A054819, so the sum of terms up to but not including the n-th one is A000720(A054819(n - 1)).

A333253 Lengths of maximal strictly increasing subsequences in the sequence of prime gaps (A001223).

Original entry on oeis.org

2, 2, 2, 3, 2, 1, 3, 1, 2, 1, 2, 3, 1, 2, 3, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 1, 3, 2, 1, 1, 1, 2, 1, 3, 1, 3, 2, 4, 1, 1, 3, 3, 2, 2, 3, 1, 3, 1, 2, 3, 2, 2, 1, 1, 3, 1, 1, 2, 1, 1, 2, 1, 3, 1, 2, 4, 2, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 3, 1, 3, 1, 3, 3, 1, 2, 2, 2

Offset: 1

Gus Wiseman, Mar 18 2020

nonn

Prime gaps are differences between adjacent prime numbers.

			The prime gaps split into the following strictly increasing subsequences: (1,2), (2,4), (2,4), (2,4,6), (2,6), (4), (2,4,6), (6), (2,6), (4), (2,6), (4,6,8), (4), (2,4), (2,4,14), ...

Wikipedia, Longest increasing subsequence

The weakly decreasing version is A333212.
The weakly increasing version is A333215.
The unequal version is A333216.
First differences of A333231 (if its first term is 0).
The strictly decreasing version is A333252.
The equal version is A333254.
Prime gaps are A001223.
Strictly increasing runs of compositions in standard order are A124768.
Positions of strict ascents in the sequence of prime gaps are A258025.
Cf. A000040, A054819, A064113, A084758, A333214, A333230, A333255.

Length/@Split[Differences[Array[Prime,100]],#1<#2&]//Most

Partial sums are A333231. The partial sum up to but not including the n-th one is A333382(n).

A052165 Primes at which the difference pattern X,2,4,2,Y (X and Y >= 6) occurs in A001223.

Original entry on oeis.org

191, 821, 2081, 3251, 9431, 13001, 15641, 18041, 18911, 25301, 31721, 34841, 51341, 62981, 67211, 69491, 72221, 77261, 81041, 82721, 97841, 99131, 109841, 116531, 119291, 122201, 135461, 157271, 171161, 187631, 194861, 201491, 217361

Offset: 1

Labos Elemer, Jan 26 2000

nonn

All terms == 11 (mod 30). - Robert Israel, Nov 30 2015

			191 is here because 191 + 2 = 193, 191 + 4 + 2 = 197, 191 + 2 + 4 + 2 = 199 are primes; the prime preceding 191 is 181; the prime following 199 is 211; and the corresponding differences are 10 and 12. Thus the d-pattern "around 191" is {10,2,4,2,12}.

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

Cf. A001223, A022008, A047078, A052160, A052162, A052163, A052164, A052167, A052168.

Primes:= select(isprime,[2,seq(i,i=3..10^6,2)]):
Gaps:= Primes[2..-1]-Primes[1..-2]:
Primes[select(t -> Gaps[t] = 2 and Gaps[t+1] = 4 and Gaps[t+2] = 2 and Gaps[t-1] >= 6 and Gaps[t+3]>=6, [$2..nops(Gaps)-3])]; # Robert Israel, Nov 30 2015

With[{x = 6, y = 6, s = Partition[#, 6, 1] &@ Prime@ Range[3*10^4]}, Select[s, And[First@ # >= x, Last@ # >= y, Most@ Rest@ # == {2, 4, 2}] &@ Differences@ # &]][[All, 2]] (* Michael De Vlieger, Oct 26 2017 *)

A118924 Primes for which the weight as defined in A117078 is 53 and the gap as defined in A001223 is 52.

Original entry on oeis.org

19609, 547171, 3099757, 3282289, 3401221, 4286851, 4648099, 5544859, 5622769, 5731207, 5868901, 6387559, 6581857, 6949147, 6985081, 7382899, 7412791, 7675141, 7697401, 8203021, 8366791, 9190411, 9649921, 9990499, 9994951

Offset: 1

Rémi Eismann, May 25 2006, May 04 2007

nonn

The prime numbers in this sequence are of the form (106i-1) with i=(level(n)+1)/2, level(n) defined in A117563.

			Prime(2226) = prime(2225) + (prime(2225) mod 53) = 19609 + (19609 mod 53) = 19661
g(n) = 19661 - 19609 = 53 - 1 = 52

Cf. A117078, A001223, A001359, A117563, A074822, A118922, A118359, A118924, A119504, A119597, A119596, A119595, A118380.

A333252 Lengths of maximal strictly decreasing subsequences in the sequence of prime gaps (A001223).

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 2, 3, 1, 1, 2, 3, 2, 1, 3, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 1, 2, 3, 1, 3, 1, 2, 2, 1, 1, 2, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 3, 2, 1, 2, 2, 2, 2, 1, 2, 2, 1, 3, 3, 1, 1, 2, 2, 1, 1, 2, 3, 2, 3, 2, 2, 2, 2, 2, 1, 3, 1, 3, 1, 2, 1

Offset: 1

Gus Wiseman, Mar 18 2020

nonn

Prime gaps are differences between adjacent prime numbers.

			The prime gaps split into the following strictly decreasing subsequences: (1), (2), (2), (4,2), (4,2), (4), (6,2), (6,4,2), (4), (6), (6,2), (6,4,2), (6,4), (6), (8,4,2), (4,2), (4), (14,4), (6,2), (10,2), (6), (6,4), (6), ...

Wikipedia, Longest increasing subsequence

The weakly decreasing version is A333212.
The weakly increasing version is A333215.
The unequal version is A333216.
First differences of A333230 (if the first term is 0).
The strictly increasing version is A333253.
The equal version is A333254.
Prime gaps are A001223.
Strictly decreasing runs of compositions in standard order are A124769.
Positions of strict descents in the sequence of prime gaps are A258026.
Cf. A000040, A064113, A084758, A124764, A124766, A258025, A333213, A333214, A333252, A333256.

Length/@Split[Differences[Array[Prime,100]],#1>#2&]//Most

Partial sums are A333230. The partial sum up to but not including the n-th one is A333381(n - 1).

A047078 Primes at which difference pattern X2Y (X and Y >= 6) occurs in A001223.

Original entry on oeis.org

29, 59, 137, 149, 179, 239, 269, 419, 431, 521, 569, 599, 659, 809, 1019, 1031, 1049, 1061, 1151, 1229, 1289, 1319, 1619, 1721, 1931, 1949, 2027, 2111, 2129, 2309, 2339, 2549, 2591, 2729, 2789, 2969, 2999, 3119, 3299, 3329, 3359, 3371, 3389, 3539, 3557

Offset: 1

Labos Elemer, Jan 26 2000

nonn

			59 is here because 59 + 2 = 61 is prime, but the difference pattern around 59 is {[53] 6 [59] 2 [61] 6 [67]}.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A001223, A022008, A052160, A052162-A052168.

Select[Prime@ Range[2, 500], Times @@ Boole@ {First@ # >= 6, #[[2]] == 2, Last@ # >= 6} == 1 &@ Differences@ Prime[# + Range[-1, 2]] &@ PrimePi@ # &] (* Michael De Vlieger, Jul 04 2016 *)

A052163 Primes at which the difference pattern X24Y (X and Y >= 6) occurs in A001223.

Original entry on oeis.org

347, 641, 1277, 1607, 2237, 2267, 2657, 3527, 3671, 3917, 4001, 4127, 4637, 4931, 4967, 5477, 5501, 6197, 8087, 8231, 8537, 8861, 9461, 10331, 10427, 11171, 11777, 12107, 12917, 13757, 13901, 14081, 14321, 14627, 17027, 18251, 19991, 20477

Offset: 1

Labos Elemer, Jan 26 2000

nonn

			641 is in the sequence because 641 + 2 = 643, 641 + 2 + 4 = 647 is prime, the prime prior to 641 is 631, the prime after 647 is 653, and the corresponding differences are 10 or 6. The d-pattern is {10,2,4,6}.

Cf. A001223, A052160, A052162-A052168, A022008, A047078.

A052164 Primes at which the difference pattern X42Y (X and Y >= 6) occurs in A001223.

Original entry on oeis.org

67, 277, 613, 1447, 1663, 1693, 1783, 2137, 2377, 2707, 2797, 3163, 3847, 4153, 5413, 5437, 5737, 6547, 7207, 7753, 8623, 9007, 9277, 9337, 10267, 11113, 11827, 12037, 12157, 12373, 14557, 16447, 17203, 18127, 18307, 18517, 19207, 20143

Offset: 1

Labos Elemer, Jan 26 2000

nonn

			67 is in the sequence because 67 + 4 = 71 and 67 + 4 + 2 = 73 are primes, the prime prior to 67 is 61, the prime after 73 is 79, and the corresponding differences are 6 and 6. The d-pattern "around 67" is {6,4,2,6}.

Cf. A001223, A051260.

A052166 Primes at which the difference pattern X424Y (X and Y >= 6) occurs in A001223.

Original entry on oeis.org

37, 223, 307, 457, 853, 877, 1087, 1297, 1423, 1993, 2683, 4513, 4783, 5227, 6823, 7873, 8287, 10453, 13687, 13873, 16183, 17383, 20743, 21313, 23053, 23557, 23623, 24103, 27733, 29017, 31387, 33343, 33613, 35527, 36007, 37987, 40423, 42013

Offset: 1

Labos Elemer, Jan 26 2000

nonn

			37 is here because 37 + 4 = 41, 37 + 4 + 2 = 43, 37 + 4 + 2 + 4 = 47 are consecutive primes and the prime preceding 37 is 31, the prime following 47 is 53, and the corresponding differences are 6 and 6. Thus the d-pattern "around 37" is {6,4,2,4}.

Cf. A001223, A052160, A052162-A052168, A022008, A047078.

okQ[n_List]:=Module[{d=Differences[n]},Take[d,{2,4}]=={4,2,4} && First[d]>5&&Last[d]>5]; Transpose[Select[ Partition[ Prime[ Range[ 4400]], 6, 1],okQ]][[2]] (* Harvey P. Dale, Jul 17 2011 *)

cp's OEIS Frontend

A089180 a(n) is the smallest number m such that d(m) = d(m+1) = ... = d(m+n), where d(k) = prime(k+1) - prime(k) (A001223).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A333212 Lengths of maximal weakly decreasing subsequences in the sequence of prime gaps (A001223).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A333253 Lengths of maximal strictly increasing subsequences in the sequence of prime gaps (A001223).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A052165 Primes at which the difference pattern X,2,4,2,Y (X and Y >= 6) occurs in A001223.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A118924 Primes for which the weight as defined in A117078 is 53 and the gap as defined in A001223 is 52.

Views

Author

Keywords

Comments

Examples

Crossrefs

A333252 Lengths of maximal strictly decreasing subsequences in the sequence of prime gaps (A001223).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A047078 Primes at which difference pattern X2Y (X and Y >= 6) occurs in A001223.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A052163 Primes at which the difference pattern X24Y (X and Y >= 6) occurs in A001223.

Views

Author

Keywords

Examples

Crossrefs

A052164 Primes at which the difference pattern X42Y (X and Y >= 6) occurs in A001223.

Views

Author

Keywords

Examples