A206044 -id:A206044 - cp's OEIS frontend

A258061 a(n) = A206044(n)/210.

1069022, 1174439, 1433930, 7315048, 8383730, 19106606, 23389398, 34089553, 38835086, 71663630, 75752521, 86624609, 101137815, 107237721, 119545490, 123036966, 129093958, 130383688, 132980893, 138881703

Offset: 1

Zak Seidov, May 18 2015

nonn

Zak Seidov, Table of n, a(n) for n = 1..230

Cf. A206044.

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

A206042 Values of the difference d for 8 primes in arithmetic progression with the minimal start sequence {11 + j*d}, j = 0 to 7.

Original entry on oeis.org

1210230, 2523780, 4788210, 10527720, 12943770, 19815600, 22935780, 28348950, 28688100, 32671170, 43443330, 47330640, 51767520, 54130440, 59806740, 60625110, 63721770, 66761940, 77811300, 80892420, 87931620, 90601140, 102994500, 108310650, 115209570, 117639480

Offset: 1

Sameen Ahmed Khan, Feb 03 2012

nonn

The computations were done without any assumptions on the form of d.

			d = 2523780 then {11 + j*d}, j = 0 to 7, is {11, 2523791, 5047571, 7571351, 10095131, 12618911, 15142691, 17666471} which is 8 primes in arithmetic progression.

Sameen Ahmed Khan, Table of n, a(n) for n = 1..210
Sameen Ahmed Khan, Primes in Geometric-Arithmetic Progression, arXiv preprint arXiv:1203.2083 [math.NT], 2012. - From _N. J. A. Sloane_, Sep 15 2012

Cf. A040976, A206037, A206038, A206039, A206040, A206041, A206043, A206044, A206045.

a = 11; t = {}; Do[If[PrimeQ[{a, a + d, a + 2*d, a + 3*d, a + 4*d, a + 5*d, a + 6*d, a + 7*d}] == {True, True, True, True, True, True, True, True},
   AppendTo[t,d]], {d, 0, 200000000}]; t
Select[Range[117640000],AllTrue[11+#*Range[0,7],PrimeQ]&] (* Harvey P. Dale, Dec 31 2021 *)

A206043 Values of the difference d for 9 primes in arithmetic progression with the minimal start sequence {11 + j*d}, j = 0 to 8.

Original entry on oeis.org

32671170, 54130440, 59806740, 145727400, 224494620, 246632190, 280723800, 301125300, 356845020, 440379870, 486229380, 601904940, 676987920, 777534660, 785544480, 789052530, 799786890, 943698210, 1535452800, 1536160080, 1760583300, 1808008020

Offset: 1

Sameen Ahmed Khan, Feb 03 2012

nonn

The computations were done without any assumptions on the form of d.

			d = 54130440 then {11, 54130451, 108260891, 162391331, 216521771, 270652211, 324782651, 378913091, 433043531} which is 9 primes in arithmetic progression.

Charles R Greathouse IV, Table of n, a(n) for n = 1..1419
Sameen Ahmed Khan, Primes in Geometric-Arithmetic Progression, arXiv preprint arXiv:1203.2083 [math.NT], 2012. - From _N. J. A. Sloane_, Sep 15 2012

Cf. A040976, A206037, A206038, A206039, A206040, A206041, A206042, A206044, A206045.

a = 11; t = {}; 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}] == {True, True, True, True, True, True, True, True, True}, AppendTo[t,d]], {d, 10^9}]; t

forstep(k=210,1e10,210,forstep(p=k+11,8*k+11,k,if(!isprime(p), next(2)));print1(k", ")) \\ Charles R Greathouse IV, Feb 09 2012

a(20) corrected by Charles R Greathouse IV, Feb 09 2012

A206040 Values of the difference d for 6 primes in arithmetic progression with the minimal start sequence {7 + j*d}, j = 0 to 5.

Original entry on oeis.org

30, 150, 930, 2760, 3450, 4980, 9150, 14190, 19380, 20040, 21240, 28080, 33930, 57660, 59070, 63600, 69120, 76710, 80340, 81450, 97380, 100920, 105960, 114750, 117420, 122340, 134250, 138540, 143670, 150090, 164580, 184470, 184620, 189690, 231360, 237060

Offset: 1

Sameen Ahmed Khan, Feb 03 2012

nonn

The computations were done without any assumptions on the form of d.

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

Sameen Ahmed Khan, Table of n, a(n) for n = 1..10000
Sameen Ahmed Khan, Primes in Geometric-Arithmetic Progression, arXiv preprint arXiv:1203.2083 [math.NT], 2012. - From _N. J. A. Sloane_, Sep 15 2012

Cf. A040976, A206037, A206038, A206039, A206041, A206042, A206043, A206044, A206045.

a = 7; t = {}; Do[If[PrimeQ[{a, a + d, a + 2*d, a + 3*d, a + 4*d, a + 5*d}] == {True, True, True, True, True, True}, AppendTo[t,d]], {d, 300000}]; t
Select[Range[250000],AllTrue[7+#*Range[0,5],PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Dec 26 2017 *)

cp's OEIS Frontend

A258061 a(n) = A206044(n)/210.

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

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A206042 Values of the difference d for 8 primes in arithmetic progression with the minimal start sequence {11 + j*d}, j = 0 to 7.

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