cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 11-17 of 17 results.

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

Views

Author

Sameen Ahmed Khan, Feb 03 2012

Keywords

Comments

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

Examples

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

Links

Crossrefs

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

Programs

  • Mathematica
    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
  • PARI
    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

Extensions

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

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

Views

Author

Bernard Schott, Jan 23 2023

Keywords

Comments

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'.

Examples

			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.

Links

Crossrefs

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).

Programs

  • PARI
    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

Formula

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

A359410 Integers d such that the longest possible arithmetic progression (AP) of primes with common difference d has exactly 6 elements.

Original entry on oeis.org

30, 60, 90, 120, 180, 240, 270, 300, 330, 360, 390, 450, 480, 510, 540, 570, 600, 660, 690, 720, 750, 780, 810, 870, 900, 930, 960, 990, 1020, 1080, 1110, 1140, 1170, 1200, 1230, 1290, 1320, 1350, 1380, 1410, 1440, 1500, 1530, 1560, 1590, 1620, 1650, 1710, 1740
Offset: 1

Views

Author

Bernard Schott, Jan 29 2023

Keywords

Comments

The 6 elements are not necessarily consecutive primes.
A342309(d) gives the first element of the smallest AP with 6 elements whose common difference is a(n) = d.
All the terms are positive multiples of 30 (A249674) but are not multiples of 7 and also must not belong to A206041; indeed, terms d' in A206041 correspond to the longest possible APs of primes that have exactly 7 elements with this common difference d'.

Examples

			d = 30 is a term because the longest possible APs of primes with common difference d = 30 all have 6 elements; the first such APs start with 7, 107, 359, .... The smallest one is (7, 37, 67, 97, 127, 157); then 187 = 11*17.
d = 60 is another term because the longest possible APs of primes with common difference d = 60 all have 6 elements; the first such APs start with 11, 53, 641, .... The smallest one is (11, 71, 131, 191, 251, 311); then 371 = 7*53.
d = 150 is not a term because the longest possible AP of primes with common difference d = 150 is (7, 157, 307, 457, 607, 757, 907) which has 7 elements; this last one is unique.

Links

Crossrefs

Cf. A123556, A342309.
Subsequence of A249674.
Longest AP of prime numbers with exactly k elements: A007921 (k=1), A359408 (k=2), A206037 (k=3), A359409 (k=4), A206039 (k=5), this sequence (k=6), A206041 (k=7), no sequence for (k=8) and (k=9), A360146 (k=10), A206045 (k=11).

Programs

  • Maple
    filter := d -> (irem(d, 30) = 0) and (irem(d, 7) <> 0) and not (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 .. 1740)]);

Formula

m is a term iff A123556(m) = 6.

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

Views

Author

Sameen Ahmed Khan, Feb 03 2012

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)

A360146 Integers d such that the longest possible arithmetic progression (AP) of primes with common difference d has exactly 10 elements.

Original entry on oeis.org

210, 420, 630, 840, 1050, 1260, 1470, 1680, 1890, 2100, 2520, 2730, 2940, 3150, 3360, 3570, 3780, 3990, 4200, 4410, 4830, 5040, 5250, 5460, 5670, 5880, 6090, 6300, 6510, 6720, 7140, 7350, 7560, 7770, 7980, 8190, 8400, 8610, 8820, 9030, 9450, 9660, 9870, 10080, 10290, 10500, 10710, 10920
Offset: 1

Views

Author

Bernard Schott, Mar 09 2023

Keywords

Comments

The 10 elements are not necessarily consecutive primes.
All the terms are positive multiples of 210 = 7# but are not multiples of 11 and also must not belong to A206045, where the first term is 1536160080; indeed, terms d' in A206045 correspond to the longest possible APs of primes that have exactly 11 elements with these common differences d'.
A342309(d) gives the first element of the smallest AP with 10 elements whose common difference is a(n) = d.

Examples

			d = 210 is a term because the longest possible APs of primes with common difference d = 210 all have 10 elements. The first such AP is (199, 409, 619, 829, 1039, 1249, 1459, 1669, 1879, 2089), then 2299 = 11*209.
d = 420 is another term because the longest possible APs of primes with common difference d = 420 all have 10 elements; the first such APs start with 52879, 3544939, ... The smallest one is (52879, 53299, 53719, 54139, 54559, 54979, 55399, 55819, 56239, 56659), then 57079 = 11*5189.

Links

Crossrefs

Cf. A123556, A173919, A342309.
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), this sequence (k=10), A206045 (k=11).

Programs

  • PARI
    A053669(n) = forprime(p=2, , if(n%p, return(p)));
f(n) = my(p=A053669(n)); for (i=1, p-1, if (!isprime(p+i*n), return(p-1))); p; \\ A123556
isok(n) = f(n) == 10; \\ Michel Marcus, Mar 10 2023

Formula

m is a term iff A123556(m) = 10.

A360735 Even 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

16, 22, 26, 32, 44, 46, 52, 56, 58, 62, 70, 74, 76, 82, 86, 88, 92, 100, 106, 112, 116, 118, 122, 128, 130, 136, 140, 142, 146, 148, 152, 158, 160, 166, 170, 172, 176, 182, 184, 194, 196, 200, 202, 206, 212, 214, 218, 224, 226, 232, 236, 242, 244, 250, 254, 256, 262, 266, 268
Offset: 1

Views

Author

Bernard Schott, Feb 19 2023

Keywords

Comments

Similar sequence with odd integers d is A040976 \ {0}.
Terms are even numbers that are not divisible by 3 and that are not also in A206037.
These longest corresponding APs are of the form (q, q+d) with q odd primes (see examples).
This subsequence of A359408 corresponds to the second case '2 is one less than prime 3' (see A173919); the first case is linked to A040976.
A342309(d) gives the first element of the smallest such AP with 2 elements whose common difference is a(n) = d.

Examples

			d = 16 is a term because the first longest APs of primes with common difference 16 are (3, 19), (7,23), (13, 29), ... and all have 2 elements because next elements should be respectively 35, 39 and 45 that are all composite; the first such AP 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), ... and all have 2 elements because next elements should be respectively 51, 63 and 75 that are all composite; the first such AP that starts with A342309(22) = 7 is (7, 29).

Links

Crossrefs

Equals A359408 \ A040976.
Cf. A047235, A123556, A206037, A342309.

Programs

  • Maple
    filter := d -> (irem(d, 2) = 0) and (irem(d, 3) <> 0) and not isprime(3+d) or isprime(3+d) and not isprime(3+2*d) : select(filter, [`$`(1 .. 270)]);
isA360735 := d -> isA047235(d) and not isA206037(d): # Peter Luschny, Mar 03 2023
  • Mathematica
    Select[Range[2, 270, 2], Mod[#, 3] > 0 && Nand @@ PrimeQ[{# + 3, 2*# + 3}] &] (* Amiram Eldar, Mar 03 2023 *)
  • PARI
    isok(d) = !(d%2) && (d%3) && !(isprime(d+3) && isprime(2*d+3)); \\ Michel Marcus, Mar 03 2023

Formula

If m is a term then A123556(m) = 2, but the converse is false: a counterexample is A123556(11) = 2 and 11 is not a term.

A240233 a(n) is the smallest prime number such that both a(n) + 6n and a(n) + 12n are prime numbers.

Original entry on oeis.org

5, 5, 5, 5, 7, 7, 5, 5, 5, 7, 5, 7, 11, 5, 11, 5, 7, 23, 13, 11, 5, 5, 41, 5, 7, 37, 29, 11, 5, 13, 7, 5, 13, 23, 13, 7, 5, 5, 23, 11, 11, 5, 5, 13, 7, 5, 29, 23, 13, 7, 5, 19, 41, 13, 17, 11, 7, 5, 19, 7, 7, 7, 5, 5, 7, 5, 7, 11, 29, 13, 5, 17, 5, 19, 7, 7, 5
Offset: 1

Views

Author

Lei Zhou, Apr 02 2014

Keywords

Comments

a(n), a(n) + 6n, and a(n) + 12n form an arithmetic progression with a common difference of 6n.
If the interval is not a multiple of six, such an arithmetic progression of primes cannot exist unless a(n)=3. For example, 3,5,7 has an interval of 2; 3,7,11 has an interval of 4; and 3,11,19 has an interval of 8, as in A115334 and A206037.
Conjecture: a(n) is defined for all n > 0.

Examples

			n=1, 6n=6. 5,11,17 are all prime numbers with an interval of 6. So a(1)=5;
...
n=13, 6n=78. 5+78=83, 5+2*78=161=7*23(x); 7+78=85(x); 11+78=89, 11+78*2=167. 11,89,167 are all prime numbers with an interval of 78. So a(13)=11.

Links

Crossrefs

Cf. A000040, A115334, A206037, A240087.

Programs

  • Mathematica
    Table[diff = n*6; k = 1; While[k++; p = Prime[k]; cp1 = p + diff; cp2 = p + 2*diff; ! ((PrimeQ[cp1]) && (PrimeQ[cp2]))]; p, {n, 77}]
Previous Showing 11-17 of 17 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.