A033168 -id:A033168 - 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

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

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

N. J. A. Sloane, Oct 17 2007

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

			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

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

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

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

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