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 1-4 of 4 results.

A327465 First differences of A309095.

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane, Sep 29 2019

Keywords

Links

Crossrefs

Cf. A309095.

A309270 a(n) is the largest k such that the first k odd primes can be covered by n arithmetic progressions of primes.

Original entry on oeis.org

3, 5, 10, 13, 18, 22, 24, 27, 31, 34, 39, 41, 45, 50, 55, 62, 64, 68, 73, 79, 81, 89, 91, 96, 99, 102, 107, 110, 115, 119, 124, 128, 133, 137, 142, 145, 151, 156, 162, 166, 170, 174, 177, 182, 185, 190, 193, 199, 203, 208
Offset: 1

Views

Author

Dmitry Kamenetsky, Jul 20 2019

Keywords

Comments

Here we allow the arithmetic progressions to contain one or more terms.
The first 1000 odd primes can be covered with 221 arithmetic progressions of primes (see Links).
Finding the smallest n for a given k is a set covering problem with a binary variable for each arithmetic progression and a constraint for each of the first k odd primes. - Rob Pratt, Aug 26 2019

Examples

			1 arithmetic progression of primes is needed to cover the first 3 odd primes: (3,5,7). So a(1) = 3. Note that we cannot cover the first 4 odd primes with 1 arithmetic progression.
2 arithmetic progressions of primes are needed to cover the first 5 odd primes: (3,7,11), (5,13). So a(2) = 5.
3 arithmetic progressions of primes are needed to cover the first 10 odd primes: (3,17,31), (5,11,17,23,29), (7,13,19). So a(3) = 10.
4 arithmetic progressions of primes are needed to cover the first 13 odd primes: (3,13,23), (5,17,29,41), (7,19,31,43), (11,37). So a(4) = 13.
5 arithmetic progressions of primes are needed to cover the first 18 odd primes: (5,11,17,23,29), (7,19,31,43), (41,47,53,59), (13,37,61), (3,67). So a(5) = 18.

Links

Crossrefs

Cf. A309095.

Extensions

a(27)-a(50) from Rob Pratt, Aug 26 2019

A327466 Number of nonempty subsets of [1..n] which are geometric progressions with rational ratio and are locally maximal.

Original entry on oeis.org

1, 1, 3, 4, 8, 13, 19, 23, 27, 36, 46, 55, 67, 80, 94, 103, 119, 132, 150, 167, 187, 208, 230, 250, 266, 291, 311, 336, 364, 393, 423, 447, 479, 512, 546
Offset: 1

Views

Author

N. J. A. Sloane, Sep 29 2019

Keywords

Comments

"Locally maximal" subsets are those subsets in geometrical progression that cannot be extended to a larger subset of [1..n] in geometric progression. [Comment made precise by Giovanni Resta, Sep 30 2019.]
One might have expected that the GP would be required to have an integer ratio, but in fact we allow rational ratios. The GPs can be assumed to be strictly increasing. - N. J. A. Sloane, Oct 03 2019

Examples

			Illustrations of some initial terms:
n=3: (12),(13),(23).
n=4: (124),(13),(23),(34).
n=8: (1248), plus all 28 pairs (ij) from [1..8] except the six subsets of (1248), so a(8) = 1 + 28 - 6 = 23.

Crossrefs

See A327469 for GPs of length > 2.
Cf. A309095.

Programs

  • Mathematica
    a[1] = 1; a[n_] := Block[{t = Select[ Subsets[ Range[n], {2, Ceiling[ Log2[n + 1]]}], Length@ Union[ Rest[#]/ Most[#]] == 1 &], i = 2}, t = Reverse@ SortBy[t, Length]; i=2; While[i <= Length[t], If[ AnyTrue[ Take[t, i-1], SubsetQ[#, t[[i]]] &], t = Delete[t, i]; i=2; Continue[], i++]]; Length@ t]; Array[a, 16] (* Giovanni Resta, Sep 30 2019 *)

Extensions

a(9)-a(35) from Giovanni Resta, Sep 30 2019

A327469 Number of nonempty subsets of [1..n] which are geometric progressions of length >= 3 with rational ratio and are locally maximal.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 1, 1, 3, 3, 3, 4, 4, 4, 4, 5, 5, 7, 7, 8, 8, 8, 8, 8, 12, 12, 12, 13, 13, 13, 13, 14, 14, 14, 14, 19, 19, 19, 19, 19, 19, 19, 19, 20, 22, 22, 22, 23, 29, 33, 33, 34, 34, 34, 34, 34, 34, 34, 34, 35, 35, 35, 37, 40, 40, 40, 40, 41, 41, 41, 41, 45, 45, 45, 49, 50, 50, 50, 50, 51, 55, 55, 55, 56, 56, 56, 56, 56, 56, 58, 58, 59, 59, 59, 59, 60, 60, 66, 68, 77
Offset: 1

Views

Author

N. J. A. Sloane, Oct 03 2019, using data computed by Rob Pratt

Keywords

Comments

Exceptionally, DATA section give first 100 terms.
One might have expected that the GP would be required to have an integer ratio, but in fact we allow rational ratios. The GPs can be assumed to be increasing.

Crossrefs

Cf. A309095, A327466.
Showing 1-4 of 4 results.

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

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