A006560 -id:A006560 - cp's OEIS frontend

A335278 First index of strictly decreasing prime quartets.

11, 18, 24, 47, 58, 62, 87, 91, 111, 114, 127, 132, 146, 150, 157, 180, 210, 223, 228, 232, 242, 259, 260, 263, 269, 274, 275, 282, 283, 284, 299, 300, 309, 321, 344, 350, 351, 363, 364, 367, 368, 369, 375, 378, 382, 388, 393, 399, 406, 409, 413, 431, 442, 446

Offset: 1

Gus Wiseman, May 30 2020

nonn

Let g(i) = prime(i + 1) - prime(i). These are numbers k such that g(k) > g(k + 1) > g(k + 2).

			The first 10 strictly decreasing prime quartets:
   31  37  41  43
   61  67  71  73
   89  97 101 103
  211 223 227 229
  271 277 281 283
  293 307 311 313
  449 457 461 463
  467 479 487 491
  607 613 617 619
  619 631 641 643
For example, 211 is the 47th prime, and the primes (211,223,227,229) have differences (12,4,2), which are strictly decreasing, so 47 is in the sequence.

Prime gaps are A001223.
Second prime gaps are A036263.
Strictly increasing prime quartets are A335277.
Equal prime quartets are A090832.
Weakly increasing prime quartets are A333383.
Weakly decreasing prime quartets are A333488.
Unequal prime quartets are A333490.
Partially unequal prime quartets are A333491.
Positions of adjacent equal prime gaps are A064113.
Positions of strict ascents in prime gaps are A258025.
Positions of strict descents in prime gaps are A258026.
Positions of adjacent unequal prime gaps are A333214.
Positions of weak ascents in prime gaps are A333230.
Positions of weak descents in prime gaps are A333231.
Indices of strictly decreasing rows of A066099 are A333256.
Lengths of maximal weakly increasing sequences of prime gaps are A333215.
Lengths of maximal strictly decreasing sequences of prime gaps are A333252.
Cf. A000040, A006560, A031217, A054800, A054804, A059044, A084758, A089180, A333253.

ReplaceList[Array[Prime,100],{_,x_,y_,z_,t_,_}/;y-x>z-y>t-z:>PrimePi[x]]

prime(a(n)) = A054804(n).

A354485 Triangle read by rows: row n gives the arithmetic progression of exactly n primes with minimal final term, cf. A354376.

Original entry on oeis.org

2, 2, 3, 3, 5, 7, 7, 19, 31, 43, 5, 11, 17, 23, 29, 7, 37, 67, 97, 127, 157, 7, 157, 307, 457, 607, 757, 907, 881, 1091, 1301, 1511, 1721, 1931, 2141, 2351, 3499, 3709, 3919, 4129, 4339, 4549, 4759, 4969, 5179, 199, 409, 619, 829, 1039, 1249, 1459, 1669, 1879, 2089

Offset: 1

Bernard Schott, May 29 2022

For the corresponding values of the first term, the last term and the common difference of these arithmetic progressions, see respectively A354377, A354376 and A354484.
Without "exactly", we get A133277.
The primes in these arithmetic progressions need not be consecutive. (The smallest prime at the start of a run of exactly n consecutive primes in arithmetic progressions is A006560(n).)

			Triangle begins:
    2;
    2,    3;
    3,    5,    7;
    7,   19,   31,   43;
    5,   11,   17,   23,   29;
    7,   37,   67,   97,  127,  157;
    7,  157,  307,  457,  607,  757,  907;
  881, 1091, 1301, 1511, 1721, 1931, 2141, 2351;
  ...

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section A5, Arithmetic progressions of primes, pp. 25-28.

Michael S. Branicky, Table of n, a(n) for n = 1..231 (using A354376 and A354377)

Cf. A006560, A133277, A354376, A354377, A354484.

T(n, 1) = A354377.

T(n, n) = A354376.

A373817 Positions of terms > 1 in the run-lengths of the first differences of the odd primes.

Original entry on oeis.org

2, 14, 34, 36, 42, 49, 66, 94, 98, 100, 107, 117, 147, 150, 169, 171, 177, 181, 199, 219, 250, 268, 315, 333, 361, 392, 398, 435, 477, 488, 520, 565, 570, 585, 592, 595, 628, 642, 660, 666, 688, 715, 744, 765, 772, 778, 829, 842, 897, 906, 931, 932, 961, 1025

Offset: 1

Gus Wiseman, Jun 23 2024

nonn

Positions of terms > 1 in A333254. In other words, the a(n)-th run of differences of odd primes has length > 1.

			Primes 54 to 57 are {251, 257, 263, 269}, with differences (6,6,6). This is the 49th run, and the first of length > 2.

Positions of adjacent equal prime gaps are A064113.
Positions of adjacent unequal prime gaps are A333214.
Positions of terms > 1 in A333254, run-lengths A373821, firsts A335406.
A000040 lists the primes, differences A001223.
A027833 gives antirun lengths of odd primes, run-lengths A373820.
A046933 counts composite numbers between primes.
A065855 counts composite numbers up to n.
A071148 gives partial sums of odd primes.
Cf. A006560, A006562, A029707, A031217, A037201, A038664, A069010, A073051, A089180, A251092.

Join@@Position[Length /@ Split[Differences[Select[Range[1000],PrimeQ]]] // Most,x_Integer?(#>1&)]

A330362 a(n) is the least number that starts an arithmetic progression of n consecutive lucky numbers.

Original entry on oeis.org

1, 1, 37, 87, 87, 72979

Offset: 1

Amiram Eldar, Dec 12 2019

The difference is 6 in the arithmetic progressions that correspond to a(3)-a(6).
Calculated using Hugo van der Sanden's Lucky numbers up to 10^9 (private communication).
a(7) > 999999990, if it exists.
a(7) > 4*10^9, if it exists. - Giovanni Resta, May 10 2020

			a(3) = 37 since {37, 43, 49} = {37, 37 + 6, 37 + 2*6} are the least 3 consecutive lucky numbers in an arithmetic progression.

Cf. A000959, A006560.

Name edited by Petros Hadjicostas, Jun 06 2020

A349642 Smallest prime such that the next n prime gaps are in arithmetic progression.

Original entry on oeis.org

2, 2, 2, 17, 347, 2903, 15373, 128981, 19641263, 245333213, 245333213, 27797667517, 68439250465123, 68439250465123

Offset: 0

Pontus von Brömssen, Nov 23 2021

Equivalently, a(n) is the smallest prime p = prime(k) such that there is a polynomial f of degree at most 2 such that f(j) = prime(j) for k <= j <= k + n.
Any sequence of at most 2 terms is considered to be a degenerate arithmetic progression, so a(n) = 2 (the smallest prime) for n <= 2.
a(n) is the smallest prime p = prime(k) such that A036263(k) = A036263(k+1) = ... = A036263(k+n-2).

			The three prime gaps following the prime 17 are 2, 4, and 6, which are in arithmetic progression. This is not true for any smaller prime, so a(3) = 17.
The eight prime gaps following the prime 19641263 are 20, 18, 16, 14, 12, 10, 8, and 6, which are in arithmetic progression. This is not true for any smaller prime, so a(8) = 19641263.

From n = 3, second row of A349644.

Cf. A001223, A006560, A036263, A348927.

a(12)-a(13) from Martin Ehrenstein, Dec 05 2021

A354484 Common differences associated with the arithmetic progressions of primes in A354376.

Original entry on oeis.org

0, 1, 2, 12, 6, 30, 150, 210, 210, 210, 30030, 13860, 60060, 420420, 4144140, 9699690, 87297210, 717777060, 4180566390, 18846497670, 26004868890

Offset: 1

Bernard Schott, May 28 2022

Equivalently: Let i, i+d, i+2d, ..., i+(n-1)d be an arithmetic progression of exactly n primes; choose the one which minimizes the last term; then a(n) = common difference d.
The word "exactly" requires both i-d and i+n*d to be nonprime; without "exactly", we get A093364.
For the corresponding values of the first term and the last term, see respectively A354377 and A354376. For the actual arithmetic progressions, see A354485.
The primes in these arithmetic progressions need not be consecutive. (The smallest prime at the start of a run of exactly n consecutive primes in arithmetic progression is A006560(n).)

			The first few corresponding arithmetic progressions are:
d = 0:   (2);
d = 1:   (2, 3);
d = 2:   (3, 5, 7);
d = 12:  (7, 19, 31, 43);
d = 6:   (5, 11, 17, 23, 29);
d = 30:  (7, 37, 67, 97, 127, 157);
d = 150: (7, 157, 307, 457, 607, 757, 907).

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section A5, Arithmetic progressions of primes, pp. 25-28.

Cf. A006560, A093364, A354376, A354377, A354485.

a(1) = 0, then for n > 1, a(n) = (A354376(n) - A354377(n)) / (n-1).

a(7)-a(21) via A354376, A354377 from Michael S. Branicky, May 28 2022

A348927 a(n) is the smallest prime followed by n prime gaps in arithmetic progression with a common difference of 2.

Original entry on oeis.org

2, 5, 17, 347, 2903, 15373, 128981, 95285633, 2426256797, 41193770837, 706748065831, 68439250465123, 68439250465123

Offset: 1

Marc Morgenegg, Nov 04 2021

a(n) is the smallest prime p for which the n+1 consecutive primes starting at p can be written as p + c*k + k^2, 0 <= k <= n, for some c >= 0. - Pontus von Brömssen, Nov 07 2021

73389327921983 is the smallest prime followed by exactly 12 such gaps. - Martin Ehrenstein, Nov 20 2021

			a(2)=5, the 2 prime gaps after 5 are 2,4.
a(3)=17, the 3 prime gaps after 17 are 2,4,6.
a(4)=347, the 4 prime gaps after 347 are 2,4,6,8.
a(5)=2903, the 5 prime gaps after 2903 are 6,8,10,12,14.

Cf. A349121 (same, but starting prime gap = 2), A001223 (prime gaps), A036263 (2nd differences), A158939 (monotonic increasing), A006560 (consecutive primes in arithmetic progression).

First column of A094749 (except first few terms).

Join[{p=2},Table[While[Union@Differences[NextPrime[p,Range[0,n]],2]!={2},p=NextPrime@p];p,{n,2,7}]] (* Giorgos Kalogeropoulos, Nov 05 2021 *)

a(8) from Giorgos Kalogeropoulos, Nov 05 2021
a(9) from Pontus von Brömssen, Nov 07 2021
a(10)-a(13) from Martin Ehrenstein, Nov 20 2021

A350005 a(n) is the smallest number that starts an arithmetic progression of n consecutive ludic numbers (A003309), or 0 if no such number exists.

Original entry on oeis.org

1, 1, 1, 71, 6392047

Offset: 1

Pontus von Brömssen, Dec 08 2021

a(n) is the smallest ludic number A003309(k), such that A260723(k) = A260723(k+1) = ... = A260723(k+n-2).

a(6) > 10^8 (unless a(6) = 0).

			The first arithmetic progression of 3 consecutive ludic numbers is (1, 2, 3), so a(3) = 1.
The first arithmetic progression of 4 consecutive ludic numbers is (71, 77, 83, 89), so a(4) = 71.
The first arithmetic progression of 5 consecutive ludic numbers is (6392047, 6392077, 6392107, 6392137, 6392167), so a(5) = 6392047.

From n = 3, first row of A350007.
Cf. A003309, A260723.
Counterparts for other sequences than ludic numbers: A006560 (primes), A228433 (abundant numbers), A231623 (deficient numbers), A276821 (Sophie Germain primes), A330362 (lucky numbers).

A081734 First and smallest sequence of 6 consecutive primes in arithmetic progression.

Original entry on oeis.org

121174811, 121174841, 121174871, 121174901, 121174931, 121174961

Offset: 1

Benoit Cloitre, Apr 06 2003

Those 6 terms were first found by Lander and Parkin in 1967.

Tanya Khovanova, Non Recursions

Cf. A006560, A058362.

A368640 a(0) = 2; for n > 0, a(n) is the smallest prime that differs from the next prime by 2n and is not part of a run of 3 or more consecutive primes in arithmetic progression, or -1 if no such prime exists.

Original entry on oeis.org

2, 11, 7, 23, 89, 139, 467, 113, 1831, 523, 887, 1129, 1669, 2477, 2971, 4297, 5591, 1327, 9551, 30593, 19333, 16141, 15683, 81463, 28229, 31907, 19609, 35617, 82073, 44293, 43331, 34061, 89689, 162143, 134513, 173359, 31397, 404597, 212701, 188029, 542603, 265621

Offset: 0

Jon E. Schoenfield, Jan 01 2024

nonn

Equivalently, a(0) = 2, and for n > 0, a(n) is the smallest prime p such that NextPrime(p) - p = 2n but neither p - PreviousPrime(p) nor NextPrime(NextPrime(p)) - NextPrime(p) is equal to 2n, or -1 if no such prime p exists.
Stated yet another way: define an array A such that A(i,j) is the smallest prime that is at the start of a run of exactly i consecutive prime gaps of size exactly j, or -1 if no such prime exists; then a(0) = 2 and, for n > 0, a(n) = A(1,2n).
Conjecture: a(n) is never -1.

			a(1) is the smallest prime that differs from the next prime by 2 and is not part of a run of 3 or more consecutive primes in arithmetic progression. (Thus a(1) is the smallest prime p that is the lesser of a twin prime pair (p,q) (cf. A001359) where neither p nor q is part of another twin prime pair.) The primes 3, 5, 7 are in arithmetic progression with difference 2, so none of those primes is a(1); the smallest prime that meets the requirements is 11, so a(1) = 11.
a(6) is the smallest prime that differs from the next prime by 12 and is not part of a run of 3 or more consecutive primes in arithmetic progression. Primes that differ from the next prime by 12 are 199, 211, 467, 509, 619, ... (cf. A031930), but 199 and 211 are part of a run of three consecutive primes in arithmetic progression (199, 211, and 223). Thus, since PreviousPrime(467) != 455 and NextPrime(479) != 491, 467 is not in a run of three or more consecutive primes in arithmetic progression, so a(6) = 467.

Cf. A000040, A000230, A001359, A006560, A031930.

a(n) >= A000230(n).

cp's OEIS Frontend

A335278 First index of strictly decreasing prime quartets.

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A354485 Triangle read by rows: row n gives the arithmetic progression of exactly n primes with minimal final term, cf. A354376.

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A373817 Positions of terms > 1 in the run-lengths of the first differences of the odd primes.

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A330362 a(n) is the least number that starts an arithmetic progression of n consecutive lucky numbers.

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A349642 Smallest prime such that the next n prime gaps are in arithmetic progression.

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A354484 Common differences associated with the arithmetic progressions of primes in A354376.

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A348927 a(n) is the smallest prime followed by n prime gaps in arithmetic progression with a common difference of 2.

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A350005 a(n) is the smallest number that starts an arithmetic progression of n consecutive ludic numbers (A003309), or 0 if no such number exists.

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A081734 First and smallest sequence of 6 consecutive primes in arithmetic progression.

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