A123556 -id:A123556 - cp's OEIS frontend

A342309 When A123556(n) is the number of primes (not necessarily consecutive) in the longest arithmetic progression of primes with common difference n, a(n) is the smallest term of the first such arithmetic progression.

2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 3, 2, 7, 2, 59, 2, 3, 2, 3, 2, 7, 2, 5, 2, 3, 2, 31, 2, 3, 2, 3, 2, 5, 2, 3, 2, 7, 2, 5, 2, 3, 2, 7, 2, 5, 2, 3, 2, 3, 2, 11, 2, 5, 2, 3, 2, 31, 2, 3, 2, 3, 2, 7, 2, 5, 2, 3, 2, 23, 2, 3, 2, 7, 2, 5, 2, 3, 2, 13, 2, 13

Offset: 1

Bernard Schott, Mar 08 2021

nonn

Inspired by problem A1880 in Diophante (see link).
a(n) is the last term of the n-th row of A124064.
About the k-tuples conjecture in A123556: let p = A053669(n), if the arithmetic progression of p elements starting at p with difference n consists of primes, then A123556(n) = p, otherwise A123556(n) = p-1. The first 21 terms of this sequence are precisely the same as A053669, then a(22) = 7 (corresponding to the arithmetic progression (7,29)) while A053669(22) = 3.

			There are only two consecutive primes (p,p+1) = (2,3), hence a(1) = 2.
(5,29,53) is the smallest arithmetic progression of 3 primes with common difference of 24, but (59,83,107,131) is the smallest arithmetic progression of 4 primes with common difference of 24 and there does not exist such an arithmetic progression with length > 4; hence, a(24) = 59.
(3,29), (5,31), (11,37) are the first three arithmetic progressions of primes with common difference of 26, the smallest term of the first arithmetic progression (3,29) is 3, and there does not exist such an arithmetic progression with length > 2, hence a(26) = 3.

Diophante, A1880. NP en PA (in French).

Cf. A053669, A123556, A124064.

a(2k+1) = 2.

A040976 a(n) = prime(n) - 2.

Original entry on oeis.org

0, 1, 3, 5, 9, 11, 15, 17, 21, 27, 29, 35, 39, 41, 45, 51, 57, 59, 65, 69, 71, 77, 81, 87, 95, 99, 101, 105, 107, 111, 125, 129, 135, 137, 147, 149, 155, 161, 165, 171, 177, 179, 189, 191, 195, 197, 209, 221, 225, 227, 231, 237, 239, 249, 255, 261

Offset: 1

Felice Russo

Numbers k such that k! reduced mod (k+2) is 1. - Benoit Cloitre, Mar 11 2002
The first a(n) numbers starting from 2 are divisible by primes up to prime(n-1). - Lekraj Beedassy, Jun 21 2006
The terms in this sequence are the cumulative sums of distances from one prime to another. For example for the distance from the first to 26th prime, 2 to 101, the cumulative sum of distances is 99, always the last prime, here 101, minus 2. - Enoch Haga, Apr 24 2006
The primes in this sequence are the initial primes of twin prime pairs. - Sebastiao Antonio da Silva, Dec 21 2008
Note that many, but not all, of these numbers satisfy x such that x^(x+1) = 1 mod (x+2). The first exception is 339. - Thomas Ordowski, Nov 27 2013
If this sequence had an infinite number of primes, the twin prime conjecture would follow. Sequence holds all primes in A001359. - John W. Nicholson, Apr 14 2014
From Bernard Schott, Feb 19 2023: (Start)
Equivalently, except for a(1)=0, all terms are odd integers d such that the longest possible arithmetic progression (AP) of primes with common difference d has only two elements.
For each term d, there exists only one such AP of primes, and this one always starts with A342309(d) = 2, so this unique AP is (2, 2+d) = (2, prime(m)) with m > 1; so, first examples are (2,3), (2,5), (2,7), (2,11), ... next elements should be respectively: 4, 8, 12, 20, ... that are all composite numbers.
Similar sequence with even common differences d is A360735.
This subsequence of A359408 corresponds to the first case: '2 is prime'; second case corresponding to the even common differences d is A360735. (End)

			a(13) = 39, because A000040(13) = 41.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
S. A. Khan, Primes in Geometric-Arithmetic Progression, arXiv preprint arXiv:1203.2083 [math.NT], 2012.
S. M. Ruiz and J. Sondow, Formulas for pi(n) and the n-th prime, arXiv:math/0210312 [math.NT], 2002-2014.
Wikipedia, Primes in arithmetic progression.
Index entries for sequences related to primes in arithmetic progressions.

Cf. A000040, A001223, A014689, A014692.
Equals A359408 \ A360735.
Cf. A123556, A342309.
First column of A086800, and of A379011, last diagonal of A090321, and of A162621.
See also irregular triangles A103728, A319148, A369497.

Filtered([1..10^2],IsPrime)-2; # Muniru A Asiru, Jan 31 2018

a040976 n = a000040 n - 2
a040976_list = map (subtract 2) a000040_list
-- Reinhard Zumkeller, Feb 22 2012

[NthPrime(n)-2: n in [1..60]]; // Vincenzo Librandi, Jan 31 2018

seq(ithprime(n)-2,n=1..100); # Muniru A Asiru, Jan 31 2018

Prime[Range[22]]-2 (* Vladimir Joseph Stephan Orlovsky, Apr 29 2008 *)

a(n)=prime(n)-2 \\ Charles R Greathouse IV, Nov 20 2012

a(n) = A000040(n) - 2 = Sum_{i=1..n-1} A001223(i).
For n > 2: A092953(a(n)) = 1. - Reinhard Zumkeller, Nov 10 2012
If m is a term then A123556(m) = 2, but the converse is false: a counterexample is A123556(16) = 2 and 16 is not a term. - Bernard Schott, Feb 19 2023
a(n) = Sum_{k = 2..floor(2n*log(n)+2)} (1-floor(A000720(k)/n)). [Ruiz and Sondow]. - Elias Alejandro Angulo Klein, Apr 09 2024

A007921 Numbers that are not the difference of two primes.

Original entry on oeis.org

7, 13, 19, 23, 25, 31, 33, 37, 43, 47, 49, 53, 55, 61, 63, 67, 73, 75, 79, 83, 85, 89, 91, 93, 97, 103, 109, 113, 115, 117, 119, 121, 123, 127, 131, 133, 139, 141, 143, 145, 151, 153, 157, 159, 163, 167, 169, 173, 175, 181, 183, 185, 187, 193

Offset: 1

R. Muller

Conjecturally, odd numbers k such that k+2 is composite.
Is this the same as A068780(2n-1) - 1? - J. Stauduhar, Aug 23 2012
A092953(a(n)) = 0. - Reinhard Zumkeller, Nov 10 2012
It seems that the sequence contains the squares of all primes except for 2 and 3. - Ivan N. Ianakiev, Aug 29 2013 [It does: For every prime p > 3, note that p^2 == 1 (mod 3), so p^2 cannot be q - r where q and r are primes. (If it were, then since p^2 is odd, q and r could not both be odd primes; r would have to be the even prime, 2, which would mean that p^2 = q - 2, so q = p^2 + 2 == 0 (mod 3), i.e., 3 would divide q, so q would not be prime -- a contradiction.) - Jon E. Schoenfield, May 03 2024]
Integers d such that A123556(d) = 1, that is, integers d such that the largest possible arithmetic progression (AP) of primes with common difference d has only one element. For each such d, the unique element of all the first largest APs with 1 element is A342309(d) = 2. - Bernard Schott, Jan 08 2023
If it exists, the least even term is > 10^12 (see 1st comment in A020483). - Bernard Schott, Jan 09 2023

F. Smarandache, Properties of Numbers, 1972. (See Smarandache odd sieve.)

T. D. Noe, Table of n, a(n) for n = 1..10000
Diophante, A1880. NP en PA (prime numbers in arithmetic progression) (in French).
C. Dumitrescu and V. Seleacu, editors, Some Notions and Questions in Number Theory, Vol. I.
F. Smarandache, Only Problems, Not Solutions!, 4th ed., 1993, Problem 94.
Wikipedia, Primes in arithmetic progression.
Index entries for sequences related to primes in arithmetic progressions.
Index entries for primes, gaps between.

Cf. A048859.
Complement of A030173. Cf. A001223.
Cf. also A005408, A010051.
Cf. A123556, A342309.
Largest AP of prime numbers with k elements: this sequence (k=1), A359408 (k=2), A206037 (k=3), A359409 (k=4), A206039 (k=5), A359410 (k=6), A206041 (k=7), A206042 (k=8), A206043 (k=9), A206044 (k=10), A206045 (k=11).

a007921 n = a007921_list !! (n-1)
a007921_list = filter ((== 0) . a010051' . (+ 2)) [1, 3 ..]
-- Reinhard Zumkeller, Jul 03 2015

filter :=  d -> irem(d, 2) <> 0 and not isprime(2+d) : select(filter, [`$`(1 .. 200)]); # Bernard Schott, Jan 08 2023

Lim=200;nn=10;seq:=Complement[Range[Lim],Union[Flatten[Differences/@Subsets[Prime[Range[nn]],{2}]]]];Until[AllTrue[seq,OddQ],nn++];seq (* James C. McMahon, May 04 2024 *)

is(n)=n%2 && !isprime(n+2) \\ On Polignac's conjecture; Charles R Greathouse IV, Jun 28 2013

from sympy import isprime
print([n for n in range(1, 200) if n%2 and not isprime(n + 2)]) # Indranil Ghosh, Jun 15 2017, after Charles R Greathouse IV

A206037 Values of the difference d for 3 primes in arithmetic progression with the minimal start sequence {3 + j*d}, j = 0 to 2.

Original entry on oeis.org

2, 4, 8, 10, 14, 20, 28, 34, 38, 40, 50, 64, 68, 80, 94, 98, 104, 110, 124, 134, 154, 164, 178, 188, 190, 208, 220, 230, 238, 248, 260, 280, 308, 314, 328, 344, 370, 418, 428, 430, 440, 454, 458, 484, 518, 544, 560, 574, 584, 610, 614, 628, 638, 640, 644, 650

Offset: 1

Sameen Ahmed Khan, Feb 03 2012

The computations were done without any assumptions on the form of d.
Numbers k such that k+3 and 2k+3 are both primes.
Equivalently, integers d such that the largest possible arithmetic progression (AP) of primes with common difference d has exactly 3 elements (see example). These 3 elements are not necessarily consecutive primes. In fact, for each term d, there exists only one such AP of primes, and this one starts always with A342309(d) = 3, so this AP is (3, 3+d, 3+2d). - Bernard Schott, Jan 15 2023

			d = 8 then {3, 3 + 1*8, 3 + 2*8} = {3, 11, 19}, which is 3 primes in arithmetic progression.

Sameen Ahmed Khan, Table of n, a(n) for n = 1..10000.
Diophante, A1880. NP en PA (prime numbers in arithmetic progression) (in French).
Sameen A. Khan, Primes in Geometric-Arithmetic Progression, arXiv preprint arXiv:1203.2083 [math.NT], 2012.
Wikipedia, Primes in arithmetic progression.
Index entries for sequences related to primes in arithmetic progressions.

Cf. A040976, A123556, A206038, A206039, A206040, A206041, A206042, A206043, A206044, A206045, A342309.

Largest AP of prime numbers with k elements: A007921 (k=1), A359408 (k=2), this sequence (k=3), A359409 (k=4), A206039 (k=5), A359410 (k=6), A206041 (k=7).

[n: n in [1..700] | IsPrime(3+n) and IsPrime(3+2*n)]; // Vincenzo Librandi, Dec 28 2015

filter := d -> isprime(3+d) and isprime(3+2*d) : select(filter, [$(1 .. 650)]); # Bernard Schott, Jan 16 2023

t={}; Do[If[PrimeQ[{3, 3 + d, 3 + 2*d}] == {True, True, True}, AppendTo[t, d]], {d, 1000}]; t
Select[Range[2,700,2],And@@PrimeQ[{3+#,3+2#}]&] (* Harvey P. Dale, Sep 25 2013 *)

for(n=1, 1e3, if(isprime(n + 3) && isprime(2*n + 3), print1(n, ", "))); \\ Altug Alkan, Dec 27 2015

a(n) = 2 * A115334(n). - Wesley Ivan Hurt, Feb 06 2014

m is a term iff A123556(m) = 3. - Bernard Schott, Jan 15 2023

A206045 Numbers d such that 11 + j*d is prime for j = 0 to 10.

Original entry on oeis.org

1536160080, 4911773580, 25104552900, 77375139660, 83516678490, 100070721660, 150365447400, 300035001630, 318652145070, 369822103350, 377344636200, 511688932650, 580028072610, 638663371710, 701534299830, 745828915650, 776625236100, 883476548850, 925639075620, 956863233690

Offset: 1

Sameen Ahmed Khan, Feb 03 2012

nonn

Original name: Values of the difference d for 11 primes in arithmetic progression with the minimal start sequence {11 + j*d}, j = 0 to 10.
The computations were done without any assumptions on the form of d. 21st term is greater than 10^12.
All terms are multiples of 210=2*3*5*7. - Zak Seidov, May 16 2015
Equivalently, integers d such that the longest possible arithmetic progression (AP) of primes with common difference d has exactly 11 elements (see example). These 11 elements are not necessarily consecutive primes. In fact, here, for each term d, there exists only one such AP of primes, and this one always starts with A342309(d) = 11, so this unique AP is (11, 11+d, 11+2d, 11+3d, 11+4d, 11+5d, 11+6d, 11+7d, 11+8d, 11+9d, 11+10d). - Bernard Schott, Mar 08 2023

			d = 4911773580 then {11, 4911773591, 9823547171, 14735320751, 19647094331, 24558867911, 29470641491, 34382415071, 39294188651, 44205962231, 49117735811} which is 11 primes in arithmetic progression.

Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 139.

Zak Seidov, Table of n, a(n) for n = 1..623.
Diophante, A1880. NP en PA (prime numbers in arithmetic progression) (in French).
Sameen Ahmed Khan, Primes in Geometric-Arithmetic Progression, arXiv preprint arXiv:1203.2083 [math.NT], 2012.
Wikipedia, Primes in arithmetic progression.
Index entries for sequences related to primes in arithmetic progressions.

Cf. A040976, A206038, A206040, A206042, A206043, A206044.
Cf. A123556, A173919.
Common differences for longest possible APs of primes with exactly k elements: A007921 (k=1), A359408 (k=2), A206037 (k=3), A359409 (k=4), A206039 (k=5), A359410 (k=6), A206041 (k=7), A360146 (k=10), this sequence (k=11).

a = 11; Do[If[PrimeQ[{a, a + d, a + 2*d, a + 3*d, a + 4*d, a + 5*d, a + 6*d, a + 7*d, a + 8*d, a + 9*d, a + 10*d}] == {True, True, True, True, True, True, True, True, True, True, True}, Print[d]], {d, 210,10^12, 210}] (* corrected by Zak Seidov, May 16 2015 *)
Select[Range[210,10^12,210],AllTrue[Range[0,10]#+11,PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 28 2016 *)

is(n)=for(j=1,10, if(!isprime(j*n+11), return(0))); 1 \\ Charles R Greathouse IV, May 18 2015

m is a term iff A123556(m) = 11. - Bernard Schott, Mar 08 2023

New name from Charles R Greathouse IV, May 18 2015

A206039 Values of the difference d for 5 primes in arithmetic progression with the minimal start sequence {5 + j*d}, j = 0 to 4.

Original entry on oeis.org

6, 12, 42, 48, 96, 126, 252, 426, 474, 594, 636, 804, 1218, 1314, 1428, 1566, 1728, 1896, 2106, 2574, 2694, 2898, 3162, 3366, 4332, 4368, 4716, 4914, 4926, 4962, 5472, 5586, 5796, 5838, 6048, 7446, 7572, 7818, 8034, 8958, 9168, 9204, 9714

Offset: 1

Sameen Ahmed Khan, Feb 03 2012

nonn

The computations were done without any assumptions on the form of d.
All terms are multiples of 6. - Zak Seidov, Jan 07 2014
Equivalently, integers d such that the largest possible arithmetic progression (AP) of primes with common difference d has exactly 5 elements (see example). These 5 elements are not necessarily consecutive primes. In fact, for each term d, there exists only one such AP of primes, and this one always starts with A342309(d) = 5, so this unique AP is (5, 5+d, 5+2d, 5+3d, 5+4d). - Bernard Schott, Jan 25 2023

			d = 12 then {5, 5 + 1*12, 5 + 2*12, 5 + 3*12, 5 + 4*12} = {5, 17, 29, 41, 53}, which is 5 primes in arithmetic progression.

Sameen Ahmed Khan, Table of n, a(n) for n = 1..10000
Diophante, A1880. NP en PA (prime numbers in arithmetic progression) (in French).
Sameen Ahmed Khan, Primes in Geometric-Arithmetic Progression, arXiv preprint arXiv:1203.2083 [math.NT], 2012. - From _N. J. A. Sloane_, Sep 15 2012
Wikipedia, Primes in arithmetic progression.
Index entries for sequences related to primes in arithmetic progressions.

Cf. A040976, A123556, A206038, A206040, A206042, A206043, A206044, A206045, A342309.

Largest AP of prime numbers with exactly k elements: A007921 (k=1), A359408 (k=2), A206037 (k=3), A359409 (k=4), this sequence (k=5), A359410 (k=6), A206041 (k=7), A360146 (k=10), A206045 (k=11).

filter := d -> isprime(5+d) and isprime(5+2*d) and isprime(5+3*d) and isprime(5+4*d) : select(filter, [$(1 .. 10000)]); # Bernard Schott, Jan 25 2023

t={}; Do[If[PrimeQ[{5, 5 + d, 5 + 2*d, 5 + 3*d, 5 +4*d}] == {True, True, True, True, True}, AppendTo[t, d]], {d, 10000}]; t
Select[Range[10000],AllTrue[5+#*Range[0,4],PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, May 09 2015 *)

m is a term iff A123556(m) = 3. - Bernard Schott, Jan 25 2023

A206041 Values of the difference d for 7 primes in arithmetic progression with the minimal start sequence {7 + j*d}, j = 0 to 6.

Original entry on oeis.org

150, 2760, 3450, 9150, 14190, 20040, 21240, 63600, 76710, 117420, 122340, 134250, 184470, 184620, 189690, 237060, 274830, 312000, 337530, 379410, 477630, 498900, 514740, 678750, 707850, 1014540, 1168530, 1180080, 1234530, 1251690, 1263480, 1523520, 1690590

Offset: 1

Sameen Ahmed Khan, Feb 03 2012

nonn

The computations were done without any assumptions on the form of d.
All terms are multiples of 30. - Zak Seidov, Jan 07 2014.
Equivalently, integers d such that the longest possible arithmetic progression (AP) of primes with common difference d has exactly 7 elements (see example). These 7 elements are not necessarily consecutive primes. In fact, for each term d, there exists only one such AP of primes, and this one always starts with A342309(d) = 7, so this unique AP is (7, 7+d, 7+2d, 7+3d, 7+4d, 7+5d, 7+6d). - Bernard Schott, Feb 12 2023

			d = 150 then {7, 7 + 1*150, 7 + 2*150, 7 + 3*150, 7 + 4*150, 7 + 5*150, + 7 + 6*150} = {7, 157, 307, 457, 607, 757, 907} which is 7 primes in arithmetic progression.

Sameen Ahmed Khan, Table of n, a(n) for n = 1..10000.
Diophante, A1880. NP en PA (prime numbers in arithmetic progression) (in French).
Sameen Ahmed Khan, Primes in Geometric-Arithmetic Progression, arXiv preprint arXiv:1203.2083 [math.NT], 2012. - From _N. J. A. Sloane_, Sep 15 2012
Wikipedia, Primes in arithmetic progression.
Index entries for sequences related to primes in arithmetic progressions.

Cf. A040976, A206038, A206040, A206042, A206043, A206044, A206045, A123556, A342309.

Longest AP of prime numbers with exactly k elements: A007921 (k=1), A359408 (k=2), A206037 (k=3), A359409 (k=4), A206039 (k=5), A359410 (k=6), this sequence (k=7), A360146 (k=10), A206045 (k=11).

filter := d -> isprime(7+d) and isprime(7+2*d) and isprime(7+3*d) and isprime(7+4*d) and isprime(7+5*d) and isprime(7+6*d): select(filter, [$(1 .. 1700000)]); # Bernard Schott, Feb 13 2023

a = 7; t = {}; Do[If[PrimeQ[{a, a + d, a + 2*d, a + 3*d, a + 4*d, a + 5*d, a + 6*d}] == {True, True, True, True, True, True, True}, AppendTo[t,d]], {d, 200000}]; t

m is a term iff A123556(m) = 7. - Bernard Schott, Feb 12 2023

A359408 Integers d such that the longest possible arithmetic progression (AP) of primes with common difference d has only two elements.

Original entry on oeis.org

1, 3, 5, 9, 11, 15, 16, 17, 21, 22, 26, 27, 29, 32, 35, 39, 41, 44, 45, 46, 51, 52, 56, 57, 58, 59, 62, 65, 69, 70, 71, 74, 76, 77, 81, 82, 86, 87, 88, 92, 95, 99, 100, 101, 105, 105, 106, 107, 111, 112, 116, 118, 122, 125, 128, 129, 130, 135, 136, 137, 140, 142, 146, 147, 148, 149, 152, 155

Offset: 1

Bernard Schott, Dec 30 2022

nonn

As '2 is prime' and also '2 is one less than prime 3' (see A173919), there exist two subsequences with k = 2 elements in these APs of primes (see examples).
1. If d is an odd term, then d is in A040976 \ {0} with d = prime(m) - 2, for some m >= 2, and, for each such d, there exists only one longest possible AP of primes, and this AP is always: (2, prime(m)) = (2, d+2), so starts with 2. This subsequence corresponds to the first case: '2 is prime'.
2. If d is an even term, then d is in A360735 and the longest corresponding APs of primes are of the form (q, q+d) with q odd primes. This subsequence corresponds to the second case '2 is one less than prime 3'.
A342309(d) gives the first element of the smallest AP with 2 elements whose common difference is a(n) = d.
The two elements of these APs are not necessarily consecutive primes.

			d = 1 is a term because the only longest AP of primes with common difference 1 is (2, 3) that has 2 elements because 4 is composite.
d = 3 is a term because the only longest AP of primes with common difference 3 is (2, 5) that has 2 elements because 8 is composite.
d = 5 is a term because the only longest AP of primes with common difference 5 is (2, 7) that has 2 elements because 12 is composite.
d = 16 is a term because the first longest APs of primes with common difference 16 are (3, 19), (7,23), (13, 29), ... that all have 2 elements; the first one that starts with A342309(16) = 3 is (3, 19).
d = 22 is a term because the first longest APs of primes with common difference 22 are (7, 29), (19, 41), (31, 53), ... that all have 2 elements; the first one that starts with A342309(22) = 7 is (7, 29).

Diophante, A1880. NP (Nombres Premiers) en PA (Progression Arithmétique) (in French).
Wikipedia, Primes in arithmetic progression.
Index entries for sequences related to primes in arithmetic progressions.

Cf. A123556, A173919, A342309.
Equals disjoint union of A040976 \ {0} and A360735.
Longest AP of prime numbers with exactly k elements: A007921 (k=1), this sequence (k=2), A206037 (k=3), A359409 (k=4), A206039 (k=5), A359410 (k=6), A206041 (k=7), A360146 (k=10), A206045 (k=11)

filter := d -> irem(d, 2) = 0 and irem(d, 3) <> 0 and not isprime(3+d) or irem(d, 2) = 0 and irem(d, 3) <> 0 and isprime(3+d) and not isprime(3+2*d) or isprime(d+2) : select(filter, [$(1 .. 155)]);

Select[Range[155], Mod[#,2]==0 && Mod[#,3]!=0 && !PrimeQ[3+#] || Mod[#,2]==0 && Mod[#,3]!=0 && PrimeQ[3+#] && !PrimeQ[3+2#] || PrimeQ[#+2] &] (* Stefano Spezia, Jan 08 2023 *)

m is a term iff A123556(m) = 2.

A173919 Numbers that are prime or one less than a prime.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 16, 17, 18, 19, 22, 23, 28, 29, 30, 31, 36, 37, 40, 41, 42, 43, 46, 47, 52, 53, 58, 59, 60, 61, 66, 67, 70, 71, 72, 73, 78, 79, 82, 83, 88, 89, 96, 97, 100, 101, 102, 103, 106, 107, 108, 109, 112, 113, 126, 127, 130, 131, 136, 137, 138, 139

Offset: 1

Giovanni Teofilatto, Mar 02 2010

Indices of the triangular numbers in A147846.
Integers k such that k or k+1 is prime. - Giovanni Teofilatto, Mar 05 2010
For a given common difference d, there always exists a longest possible arithmetic progression (AP) of primes, and the number of elements k in this AP of primes is necessarily a term of this sequence. See A123556 for explanations. - Bernard Schott, Mar 18 2023

Harvey P. Dale, Table of n, a(n) for n = 1..1000.
Diophante, A1880. NP (Nombres Premiers) en PA (Progression Arithmétique) (in French).
Wikipedia, Primes in arithmetic progression.
Index entries for sequences related to primes in arithmetic progressions.

Cf. A000217, A147846.
Cf. A040976, A123556, A360735.
Complement to A068780. [Giovanni Teofilatto, Mar 11 2010]

{#-1,#}&/@Prime[Range[40]]//Flatten//Union (* Harvey P. Dale, Dec 21 2018 *)

A006093 U A000040.

Definition corrected and sequence extended by R. J. Mathar, Feb 24 2010, Mar 05 2010

A359409 Integers d such that the largest possible arithmetic progression (AP) of primes with common difference d has exactly four elements.

Original entry on oeis.org

18, 24, 36, 54, 66, 72, 78, 84, 102, 108, 114, 132, 138, 144, 156, 162, 168, 174, 186, 192, 198, 204, 216, 222, 228, 234, 246, 258, 264, 276, 282, 288, 294, 306, 312, 318, 324, 336, 342, 348, 354, 366, 372, 378, 384, 396, 402, 408, 414, 432, 438, 444, 456, 462, 468, 486

Offset: 1

Bernard Schott, Jan 23 2023

nonn

These 4 elements are not necessarily consecutive primes.
A342309(d) gives the first element of the smallest AP with 4 elements whose common difference is a(n) = d.
All the terms are multiples of 6 (A008588) but are not multiples of 5 and also must not belong to A206039; indeed, terms d' in A206039 correspond to the largest possible arithmetic progression (AP) of primes that have exactly five elements with this common difference d'.

			d = 18 is a term because the largest possible APs of primes with common difference d = 18 have all 4 elements; the first such APs start with 5, 43, 53, ... The smallest one is (5, 23, 41, 59) then 77 is composite.
d = 24 is another term because the largest possible APs of primes with common difference d = 24 have all 4 elements; the first such APs start with 59, 79, 349, ... The smallest one is (59, 83, 107, 131) then 155 is composite.

Diophante, A1880. NP en PA (prime numbers in arithmetic progression) (in French).
Wikipedia, Primes in arithmetic progression.
Index entries for sequences related to primes in arithmetic progressions.

Cf. A123556, A342309.
Subsequence of A008588.
Largest AP of prime numbers with k elements: A007921 (k=1), A359408 (k=2), A206037 (k=3), this sequence (k=4), A206039 (k=5), A359410 (k=6), A206041 (k=7).

isok(d) = (d%5) && !(d%6) && !(isprime(5+d) && isprime(5+2*d) && isprime(5+3*d) && isprime(5+4*d)); \\ Michel Marcus, Jan 23 2023

m is a term iff A123556(m) = 4.

cp's OEIS Frontend

A342309 When A123556(n) is the number of primes (not necessarily consecutive) in the longest arithmetic progression of primes with common difference n, a(n) is the smallest term of the first such arithmetic progression.

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A040976 a(n) = prime(n) - 2.

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A007921 Numbers that are not the difference of two primes.

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A206037 Values of the difference d for 3 primes in arithmetic progression with the minimal start sequence {3 + j*d}, j = 0 to 2.

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A206045 Numbers d such that 11 + j*d is prime for j = 0 to 10.

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A206039 Values of the difference d for 5 primes in arithmetic progression with the minimal start sequence {5 + j*d}, j = 0 to 4.

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