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.

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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.

A235393 Integers k such that 11 + 210*k*i is prime for i = 0..7.

Original entry on oeis.org

5763, 12018, 22801, 50132, 61637, 94360, 109218, 134995, 136610, 155577, 206873, 225384, 246512, 257764, 284794, 288691, 303437, 317914, 370530, 385202, 418722, 431434, 490450, 515765, 548617, 560188, 597275, 693940, 704899, 768433, 786934, 819640, 840480
Offset: 1

Views

Author

Zak Seidov, Jan 09 2014

Keywords

Comments

Primes in AP. Longest AP with a(1) = 11 has 11 terms, see A206045.

Links

Crossrefs

Cf. A206042, A206045.

Programs

  • PARI
    is_A235393(k)=!for(i=1,7,ispseudoprime(11+210*k*i)||return)
  • PARI
    for(k=1,1e6,is_A235393(k) && print1(k",")) \\ (End)

Formula

a(n) = A206042(n)/210.

A383134 Array read by ascending antidiagonals: A(n,k) is the length of the arithmetic progression of only primes having difference n and first term prime(k).

Original entry on oeis.org

2, 1, 1, 2, 3, 1, 1, 1, 2, 1, 2, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 5, 1, 1, 1, 2, 1, 2, 3, 1, 3, 1, 2, 1, 1, 1, 1, 1, 2, 1, 4, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1
Offset: 1

Views

Author

Stefano Spezia, Apr 17 2025

Keywords

Examples

			The array begins as:
  2, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
  1, 3, 2, 1, 2, 1, 2, 1, 1, 2, ...
  2, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
  1, 3, 1, 2, 1, 2, 1, 2, 1, 1, ...
  2, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
  1, 1, 5, 3, 4, 2, 3, 1, 2, 1, ...
  1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
  1, 3, 2, 1, 2, 1, 1, 1, 2, 2, ...
  2, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
  1, 3, 1, 2, 1, 2, 1, 2, 1, 1, ...
  ...
A(2,2) = 3 since 3 primes are in arithmetic progression with a difference of 2 and the first term equal to the 2nd prime: 3, 5, and 7.
A(6,3) = 5 since 5 primes are in arithmetic progression with a difference of 6 and the first term equal to the 3rd prime: 5, 11, 17, 23, and 29.

References

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

Crossrefs

Cf. A000012, A006512, A040976, A054977, A175191 (n=2).
Cf. A088430, A206045, A237453.

Programs

  • Mathematica
    A[n_,k_]:=Module[{count=1,sum=Prime[k]},While[PrimeQ[sum+=n], count++]; count]; Table[A[n-k+1,k],{n,13},{k,n}]//Flatten

Formula

A(A006512(n),k) = 1 for n > 1.
A(A040976(n),k) = A054977(k+1).
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