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.

Showing 1-7 of 7 results.

A113872 Last term of n-term arithmetic progression of primes described in A033188 and A033189.

Original entry on oeis.org

2, 3, 7, 23, 29, 157, 907, 1669, 1879, 2089, 60881279, 147692870693, 15293983, 834172688773, 894476586329191, 1275290173878841, 259268969935081, 1027994118842320951, 1424014323186726053, 1424014323196425743, 28112131522925191409
Offset: 1

Views

Author

N. J. A. Sloane, Jan 26 2006

Keywords

Comments

The entries shown have been proved to be correct.
See A033188 and A033189 for further information.

Links

Formula

a(n) = A033189(n) + (n-1)*A033188(n). - Jens Kruse Andersen, Jun 14 2014

Extensions

a(19)-a(21) found by Jaroslaw Wroblewski, added by Jens Kruse Andersen, Jun 14 2014

A005115 Let i, i+d, i+2d, ..., i+(n-1)d be an n-term arithmetic progression of primes; choose the one which minimizes the last term; then a(n) = last term i+(n-1)d.

Original entry on oeis.org

2, 3, 7, 23, 29, 157, 907, 1669, 1879, 2089, 249037, 262897, 725663, 36850999, 173471351, 198793279, 4827507229, 17010526363, 83547839407, 572945039351, 6269243827111
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

In other words, smallest prime which is at the end of an arithmetic progression of n primes.
For the corresponding values of the first term and the common difference, see A113827 and A093364. For the actual arithmetic progressions, see A133277.
One may also minimize the common difference: this leads to A033189, A033188 and A113872.
One may also specify that the first term is the n-th prime and then minimize the common difference (or, equally, the last term): this leads to A088430 and A113834.
One may also ask for n consecutive primes in arithmetic progression: this gives A006560.

Examples

			n, AP, last term
1 2 2
2 2+j 3
3 3+2j 7
4 5+6j 23
5 5+6j 29
6 7+30j 157
7 7+150j 907
8 199+210j 1669
9 199+210j 1879
10 199+210j 2089
11 110437+13860j 249037
12 110437+13860j 262897
..........................
a(11)=249037 since 110437,124297,...,235177,249037 is an arithmetic progression of 11 primes ending with 249037 and it is the least number with this property.

References

  • R. K. Guy, Unsolved Problems in Number Theory, A5.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

For the associated gaps, see A093364. For the initial terms, see A113827. For the arithmetic progressions, see A133277.
Cf. A006560, A096003, A113830-A113834, A088430.

Programs

  • Mathematica
    (* This program will generate the 4 to 12 terms to use a[n_] to generate term 13 or higher, it will have a prolonged run time. *) a[n_] := Module[{i, p, found, j, df, k}, i = 1; While[i++; p = Prime[i]; found = 0; j = 0; While[j++; df = 6*j; (p > ((n - 1)*df)) && (found == 0), found = 1; Do[If[! PrimeQ[p - k*df], found = 0], {k, 1, n - 1}]]; found == 0]; p]; Table[a[i], {i, 4, 12}]

Formula

Green & Tao prove that this sequence is infinite, and further a(n) < 2^2^2^2^2^2^2^2^O(n). Granville conjectures that a(n) <= n! + 1 for n >= 3 and give a heuristic suggesting a(n) is around (exp(1-gamma) n/2)^(n/2). - Charles R Greathouse IV, Feb 26 2013

Extensions

a(11)-a(13) from Michael Somos, Mar 14 2004
a(14) and corrected version of a(7) from Hugo Pfoertner, Apr 27 2004
a(15)-a(17) from Don Reble, Apr 27 2004
a(18)-a(21) from Granville's paper, Jan 26 2006
Entry revised by N. J. A. Sloane, Jan 26 2006, Oct 17 2007

A204189 Benoît Perichon's 26 primes in arithmetic progression.

Original entry on oeis.org

43142746595714191, 48425980631694091, 53709214667673991, 58992448703653891, 64275682739633791, 69558916775613691, 74842150811593591, 80125384847573491, 85408618883553391, 90691852919533291, 95975086955513191, 101258320991493091, 106541555027472991, 111824789063452891, 117108023099432791, 122391257135412691, 127674491171392591, 132957725207372491, 138240959243352391, 143524193279332291, 148807427315312191, 154090661351292091, 159373895387271991, 164657129423251891, 169940363459231791, 175223597495211691
Offset: 1

Views

Author

Jonathan Sondow, Jan 14 2012

Keywords

Comments

Longest known arithmetic progression of primes as of Jan 14, 2012.
Discovered on Apr 12 2010 by Benoît Perichon using software by Jaroslaw Wroblewski and Geoff Reynolds in a distributed PrimeGrid project.

References

  • R. K. Guy, Unsolved Problems in Number Theory, 2nd ed., Springer-Verlag, 1994, A5 and A6.
  • P. Ribenboim, The New Book of Prime Number Records, Springer-Verlag, 1989, p. 224.

Links

Crossrefs

Cf. A002110, A033188, A033189, A033290, A260751, A261140, A327760, A363980, A374949.

Programs

  • Mathematica
    a[1] := 43142746595714191; a[n_] := a[n] = a[n - 1] + 5283234035979900; Table[a[n], {n, 26}] (* Alonso del Arte, Jan 14 2012 *)
Range[ 43142746595714191, 175223597495211691, 5283234035979900] (* Michael Somos, Jan 15 2012 *)
  • PARI
    a(n)=5283234035979900*n+37859512559734291 \\ Charles R Greathouse IV, Jan 15 2012

Formula

a(n) = 43142746595714191 + 5283234035979900*(n-1) for n = 1, 2, ..., 26.
a(n) = 43142746595714191 + 23681770*23#*(n-1) for n = 1..26, where 23# = 2*3*5*7*11*13*17*19*23 = 223092870 = A002110(9).

A033188 Minimal difference of any increasing arithmetic progression of n primes.

Original entry on oeis.org

0, 1, 2, 6, 6, 30, 150, 210, 210, 210, 2310, 2310, 30030, 30030, 30030, 30030, 510510, 510510, 9699690, 9699690, 9699690
Offset: 1

Views

Author

Eric W. Weisstein

Keywords

Comments

The entries shown have been proved to be correct.

Links

Crossrefs

Cf. A033189 (initial terms), A113872 (last terms).
Cf. A034386, A204189.

Formula

David W. Wilson conjectures that a(n) = n# (n primorial, A034386) for n >= 8. If this is true then a(22) onwards are: 9699690, 223092870, 223092870, 223092870, 223092870, 223092870, 223092870, 6469693230, 6469693230, ...

Extensions

Entry revised Feb 02 2006 by N. J. A. Sloane, based on suggestions from Jens Kruse Andersen.

A133276 Triangle read by rows: row n gives the first arithmetic progression of n primes with minimal distance, cf. A033188.

Original entry on oeis.org

2, 2, 3, 3, 5, 7, 5, 11, 17, 23, 5, 11, 17, 23, 29, 7, 37, 67, 97, 127, 157, 7, 157, 307, 457, 607, 757, 907, 199, 409, 619, 829, 1039, 1249, 1459, 1669, 199, 409, 619, 829, 1039, 1249, 1459, 1669, 1879, 199, 409, 619, 829, 1039, 1249, 1459, 1669, 1879, 2089, 60858179, 60860489, 60862799, 60865109, 60867419, 60869729, 60872039, 60874349, 60876659, 60878969, 60881279
Offset: 1

Views

Author

N. J. A. Sloane, Oct 17 2007

Keywords

Comments

The first 10 rows (i.e., 55 terms) are the same as for A133277 (where the final term is minimal), but here a(56) = T(11,1) = 608581797 while A133277(11,1) = 110437. - M. F. Hasler, Jan 02 2020

Examples

			Triangle begins:
    2
    2   3
    3   5   7
    5  11  17  23
    5  11  17  23   29
    7  37  67  97  127  157
    7 157 307 457  607  757  907
  199 409 619 829 1039 1249 1459 1669
  199 409 619 829 1039 1249 1459 1669 1879
  199 409 619 829 1039 1249 1459 1669 1879 2089
  ...
Row 10 is the same as in A086786, A113470, A133277, and listed as A033168. - _M. F. Hasler_, Jan 02 2020

Links

Crossrefs

For common differences see A033188, for initial terms see A033189.
Different from A133277 (from T(11,1) = a(56) on).
Cf. A086786, A113470, A033168.

Programs

  • Maple
    AP:=proc(i,d,l) [seq(i + (j-1)*d, j=1..l )]; end;

A354743 Smallest first term of arithmetic progression of exactly n primes with difference A033188(n).

Original entry on oeis.org

2, 2, 3, 41, 5, 7, 7, 881, 3499, 199, 60858179, 147692845283, 14933623, 834172298383, 894476585908771, 1275290173428391, 259268961766921, 1027994118833642281
Offset: 1

Views

Author

Bernard Schott, Jun 05 2022

Keywords

Comments

Equivalently: Let i, i+d, i+2d, ..., i+(n-1)d be an arithmetic progression of exactly n primes; choose the first one which minimizes the common difference d; then a(n) = i.
The word "exactly" requires both i-d and i+n*d to be nonprime.
For the corresponding values of the last term, see A354744.
Without "exactly", we get A033189.
The primes in these arithmetic progressions need not be consecutive.
a(n) != A033189(n) for n = 4, 8, 9, 19 because in these particular cases A113872(n) + A033188(n) is prime.
a(8) = 881 and a(9) = 3499 found by Michael S. Branicky come from A354377.
a(19) > A033189(19) = 1424014323012131633 is not known, it is the smallest first term of an arithmetic progression of exactly 19 primes with a common difference d = 9699690; then a(20) = 1424014323012131633 and a(21) = 28112131522731197609.

Examples

			The first few corresponding arithmetic progressions are:
n = 1 and d = 0:   (2);
n = 2 and d = 1:   (2, 3);
n = 3 and d = 2:   (3, 5, 7);
n = 4 and d = 6:   (41, 47, 53, 59);
n = 5 and d = 6:   (5, 11, 17, 23, 29);
n = 6 and d = 30:  (7, 37, 67, 97, 127, 157);
n = 7 and d = 150: (7, 157, 307, 457, 607, 757, 907);
n = 8 and d = 210: (881, 1091,1301, 1511, 1721, 1931, 2141, 2351).

References

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

Links

Crossrefs

Cf. A033188, A033189, A113872, A354377, A354744.

A354744 Last term of arithmetic progression of exactly n primes with difference A033188(n) and first term = A354743(n).

Original entry on oeis.org

2, 3, 7, 59, 29, 157, 907, 2351, 5179, 2089, 60881279, 147692870693, 15293983, 834172688773, 894476586329191, 1275290173878841, 259268969935081, 1027994118842320951
Offset: 1

Views

Author

Bernard Schott, Jun 05 2022

Keywords

Comments

Equivalently: Let i, i+d, i+2d, ..., i+(n-1)d be an arithmetic progression of exactly n primes; choose the first one which minimizes the common difference d; then a(n) = i+(n-1)d.
The word "exactly" requires both i-d and i+n*d to be nonprime.
Without "exactly", we get A113872.
The primes in these arithmetic progressions need not be consecutive.
a(n) != 113872(n) for n = 4, 8, 9, 19 because in these particular cases A113872(n) + A033188(n) is prime.
a(8) = 2351 and a(9) = 5179, found by Michael S. Branicky come from A354376.
a(19) > A113872(19) = 1424014323186726053 is not known, it is the last term of the arithmetic progression of exactly 19 primes with a common difference d = 9699690 and first term = A354743(19); then a(20) = 1424014323196425743 and a(21) = 28112131522925191409.

Examples

			The first few corresponding arithmetic progressions are:
n = 1 and d = 0:   (2);
n = 2 and d = 1:   (2, 3);
n = 3 and d = 2:   (3, 5, 7);
n = 4 and d = 6:   (41, 47, 53, 59);
n = 5 and d = 6:   (5, 11, 17, 23, 29);
n = 6 and d = 30:  (7, 37, 67, 97, 127, 157);
n = 7 and d = 150: (7, 157, 307, 457, 607, 757, 907);
n = 8 and d = 210: (881, 1091,1301, 1511, 1721, 1931, 2141, 2351).

References

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

Links

Crossrefs

Cf. A033188, A033189, A113872, A354377, A354743.
Showing 1-7 of 7 results.

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

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