A133276 -id:A133276 - cp's OEIS frontend

A086786 Triangle read by rows: n-th row is the smallest set of n numbers in arithmetic progression with the same prime signature.

1, 2, 3, 3, 5, 7, 5, 11, 17, 23, 5, 11, 17, 23, 29, 7, 37, 67, 97, 127, 157, 481, 485, 489, 493, 497, 501, 505, 635, 707, 779, 851, 923, 995, 1067, 1139, 635, 707, 779, 851, 923, 995, 1067, 1139, 1211, 199, 409, 619, 829, 1039, 1249, 1459, 1669, 1879, 2089, 3841

Offset: 1

Amarnath Murthy, Sep 02 2003

In this sequence "smallest" means that the last term of the arithmetic progression is minimized. A113460 minimizes the first term.

			Triangle begins:
1
2,3
3,5,7
5,11,17,23
5,11,17,23,29
7,37,67,97,127,157
481,485,489,493,497,501,505
635,707,779,851,923,995,1067,1139
635,707,779,851,923,995,1067,1139,1211
199,409,619,829,1039,1249,1459,1669,1879,2089
...

Index entries for sequences related to primes in arithmetic progressions

A087310 contains the corresponding common differences, A087308 the initial terms, A087309 the final terms.
Cf. A087308, A087309, A087310, A114660, A114663, A114670.
Row 10 is the same for A113470, A133276, A133277 and listed as A033168.

Edited and extended by David Wasserman, Jan 08 2006

Further edits by N. J. A. Sloane, Oct 17 2007

A133277 Triangle read by rows: row n gives the arithmetic progression of n primes with minimal final term, cf. A005115.

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, 110437, 124297, 138157, 152017, 165877, 179737, 193597, 207457, 221317, 235177, 249037

Offset: 1

N. J. A. Sloane, Oct 17 2007

The first 10 rows (i.e., 55 terms) are the same as for A133276 (where the common distance is minimal), but here T(11,1) = a(56) = 110437 while A133276(11,1) = 60858179. - M. F. Hasler, Jan 02 2020

For any prime p there is a p-AP (arithmetic progression of p primes) starting with p, where the common distance is given by A088430. For n between prime(k-1) and prime(k), there may be an n-AP starting at prime(k) (but not earlier) with a smaller common distance, given in A061558. - M. F. Hasler, Sep 17 2024

			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;
  ...

For common differences, see A093364. For initial terms, see A113827. For final terms, see A005115.
Differs from A133276 (from T(11,1) = a(56) on).
See also A061558 (distance in earliest n-AP), A088430 (same for primes), A231017 (second term in p-AP starting with p), A061558 (distance of n-AP starting at the smallest possible prime).

A-numbers in the Name and Crossrefs sections corrected by Bobby Jacobs, Dec 10 2016

Name edited by M. F. Hasler, Jan 02 2020

A113470 Triangle read by rows: n-th row is the smallest set of n numbers in arithmetic progression with the same number of divisors.

Original entry on oeis.org

1, 2, 3, 3, 5, 7, 5, 11, 17, 23, 5, 11, 17, 23, 29, 7, 37, 67, 97, 127, 157, 35, 65, 95, 125, 155, 185, 215, 635, 707, 779, 851, 923, 995, 1067, 1139, 635, 707, 779, 851, 923, 995, 1067, 1139, 1211, 199, 409, 619, 829, 1039, 1249, 1459, 1669, 1879, 2089, 3841, 3973

Offset: 1

David Wasserman, Jan 08 2006

In this sequence "smallest" means that the last term of the arithmetic progression is minimized and if there is still a choice then we minimize the common difference of the arithmetic progression.

			From _M. F. Hasler_, Jan 02 2020: (Start)
The triangle starts
    n | row n
   ---+------------
    1 |    1,
    2 |    2,    3,
    3 |    3,    5,    7,
    4 |    5,   11,   17,   23,
    5 |    5,   11,   17,   23,   29,
    6 |    7,   37,   67,   97,  127,  157,
    7 |   35,   65,   95,  125,  155,  185,  215,
    8 |  635,  707,  779,  851,  923,  995, 1067, 1139,
    9 |  635,  707,  779,  851,  923,  995, 1067, 1139, 1211,
   10 |  199,  409,  619,  829, 1039, 1249, 1459, 1669, 1879, 2089,
   11 | 3841, 3973,  ...
Most rows so far consist of primes with 2 divisors, rows 7, 8, 9 and 11 have squarefree semiprimes with 4 divisors.
Row 10 is A033168; also row 10 of A086786, A133276 and A133277. (End)

OEIS wiki, Primes in arithmetic progression.

Cf. A086786, A090547, A090548, A090549.

T(n,k) = A090547(n) + (k-1)*A090549(n). - R. J. Mathar, May 11 2007

A033168 Longest arithmetic progression of primes with difference 210 and minimal initial term.

Original entry on oeis.org

199, 409, 619, 829, 1039, 1249, 1459, 1669, 1879, 2089

Offset: 0

Manuel Valdivia, Apr 22 1998

Since 210 == 1 (mod 11), a progression of primes with difference 210 can't have more than ten terms because there is exactly one multiple of 11 within each run of eleven consecutive terms. For example, 2089 + 210 = 2299 = 11^2 * 19. - Alonso del Arte, Dec 22 2017, edited by M. F. Hasler, Jan 02 2020

After 199, the next prime which starts a CPAP-10 with common gap 210 is 243051733. See A094220 for further starting points. - M. F. Hasler, Jan 02 2020

Paul Glendinning, Math in Minutes: 200 Key Concepts Explained in an Instant. New York, London: Quercus (2013): pp. 316-317.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 143.

Tanya Khovanova, Non Recursions.
OEIS wiki, Primes in arithmetic progression.
Index entries for sequences related to primes in arithmetic progressions.

Row 10 of A086786, A113470, A133276, A133277.

Cf. A094220.

199 + 210*Range[0, 9] (* Paolo Xausa, Sep 14 2024 *)

forprime(p=1,,for(i=1,9,isprime(p+i*210)||next(2)); return([p+d|d<-[0..9]*210])) \\ M. F. Hasler, Jan 02 2020

a(n) = a(0) + n*210 for 0 <= n <= 9. - M. F. Hasler, Jan 02 2020

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A086786 Triangle read by rows: n-th row is the smallest set of n numbers in arithmetic progression with the same prime signature.

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A133277 Triangle read by rows: row n gives the arithmetic progression of n primes with minimal final term, cf. A005115.

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A113470 Triangle read by rows: n-th row is the smallest set of n numbers in arithmetic progression with the same number of divisors.

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A033168 Longest arithmetic progression of primes with difference 210 and minimal initial term.

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