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.

Previous Showing 11-18 of 18 results.

A169853 EKG sequence started at 11 instead of 2.

Original entry on oeis.org

11, 22, 2, 4, 6, 3, 9, 12, 8, 10, 5, 15, 18, 14, 7, 21, 24, 16, 20, 25, 30, 26, 13, 39, 27, 33, 36, 28, 32, 34, 17, 51, 42, 35, 40, 38, 19, 57, 45, 48, 44, 46, 23, 69, 54, 50, 52, 56, 49, 63, 60, 55, 65, 70, 58, 29, 87, 66, 62, 31, 93, 72, 64, 68, 74, 37, 111, 75, 78, 76, 80, 82, 41
Offset: 1

Views

Author

T. D. Noe and N. J. A. Sloane, Jun 02 2010

Keywords

Comments

A generalization of A064413.

Links

Crossrefs

For other initial terms, see A064413, A169837, A169839, A169841, A169843, A169845, A169847, A169849, A169851, A169855.

A169857 Consider the EKG sequence (A064413) started at n instead of 2; a(n) = number of steps before sequence merges with A064413, or 0 if the two sequences never merge.

Original entry on oeis.org

0, 44, 3, 44, 44, 44, 8, 44, 44, 48, 44, 44, 44, 18, 17, 58, 18, 51, 18, 44, 48, 62, 48, 44, 47, 44, 48, 82, 48, 90, 31, 48, 65, 44, 48, 101, 71, 47, 48, 116, 48, 129, 41, 39, 87, 135, 48, 65, 46, 65, 47, 148, 48, 104, 58, 71, 105, 168, 104, 187, 121, 69, 64, 69
Offset: 2

Views

Author

T. D. Noe and N. J. A. Sloane, Jun 02 2010

Keywords

Comments

For the purpose of this sequence, ignore the initial 1 in A064413 and imagine that it begins with the 2.
The upper points appear to occur at primes, the points just below the upper points at twice primes.
Are there any zeros?

Links

Crossrefs

Cf. A064413, A064421, A169837-A169856, A119415.

A255524 Let EKG-n denote the EKG sequence (A064413) started with n rather than 2, and suppose EKG-n first merges with some other EKG-i (i >= 2) sequence after f(n) (= A255583(n)) steps; then a(n) = smallest value of i such that EKG-i meets EKG-n after f(n) steps.

Original entry on oeis.org

4, 6, 2, 3, 3, 3, 2, 3, 3
Offset: 2

Views

Author

Gordon Hamilton, Feb 24 2015

Keywords

Comments

Does a(n) always exist?
See video for explanation.
Recommended for elementary school teachers to experiment with to teach factoring.

Examples

			a(5) = 3 because the EKG sequence starting with 5 (EKG-5) starts coinciding with sequences EKG-3, EKG-6, EKG-9 and EKG-12 simultaneously (when all sequences hit 18).
EKG-3:  3, 6, 2, 4, 8, 10, 5, 15, 9, 12, 14, 7, 21, 18, 16, 20, 22, 11...
EKG-6:  6, 2, 4, 8, 10, 5, 15, 3, 9, 12, 14, 7, 21, 18, 16, 20, 22, 11...
EKG-9:  9, 3, 6, 2, 4, 8, 10, 5, 15, 12, 14, 7, 21, 18, 16, 20, 22, 11...
EKG-12: 12, 2, 4, 6, 3, 9, 15, 5, 10, 8, 14, 7, 21, 18, 16, 20, 22, 11...
EKG-5:  5, 10, 2, 4, 6, 3, 9, 12, 8, 14, 7, 21, 15, 18, 16, 20, 22, 11...
Of these, the smallest EKG sequence is numbered 3 so a(5) = 3.

Links

Crossrefs

A255198 records the number of closest neighbors.
For examples of EKG-n, see A064413, A169841, A169837, A169843, A169855, A169849.
Cf. A255583.

A255583 Let EKG-n denote the EKG sequence (A064413) started with n rather than 2, and suppose EKG-n first merges with some other EKG-i (i >= 2). Then a(n) = number of steps for this to happen.

Original entry on oeis.org

3, 9, 3, 14, 9, 20, 8, 10, 17, 36, 11, 37, 21, 12, 17, 57, 13, 51, 12, 21, 39, 62, 23, 38, 39, 25, 27, 82, 23, 90, 31, 39, 49, 30, 31, 101, 66, 39, 27, 116, 31, 129, 41, 39, 66, 135, 41, 65, 46, 45, 45, 148, 46, 67, 57, 45, 83, 168, 53, 178, 91, 69, 64, 64, 53
Offset: 2

Views

Author

Gordon Hamilton, Feb 26 2015

Keywords

Comments

Merging means that the sequences are identical for all future steps. EKG-2 and EKG-5 merge at step 44. From then on the sequences are identical.

Examples

			a(5) = 14 because the EKG sequence starting with 5 (EKG-5, A169841) merges with sequences EKG-3, EKG-6, EKG-9 and EKG-12 simultaneously when all sequences hit 18.
EKG-3:  3, 6, 2, 4, 8, 10, 5, 15, 9, 12, 14, 7, 21, 18, 16, 20, 22, 11, ... (A169837)
EKG-6:  6, 2, 4, 8, 10, 5, 15, 3, 9, 12, 14, 7, 21, 18, 16, 20, 22, 11, ... (A169843)
EKG-9:  9, 3, 6, 2, 4, 8, 10, 5, 15, 12, 14, 7, 21, 18, 16, 20, 22, 11, ... (A169849)
EKG-12: 12, 2, 4, 6, 3, 9, 15, 5, 10, 8, 14, 7, 21, 18, 16, 20, 22, 11, ... (A169855)
EKG-5:  5, 10, 2, 4, 6, 3, 9, 12, 8, 14, 7, 21, 15, 18, 16, 20, 22, 11, ... (A169841)

Links

Crossrefs

Cf. A064413, A169837, A169841, A169843, A169849, A169857.
A255524 gives the smallest closest neighbor.

Programs

  • PARI
    \\ See Links section.

Extensions

More terms from Rémy Sigrist, Oct 06 2018

A339671 a(1) = 1, a(2) = 2; for n>2, a(n) = smallest number not already used that shares a prime factor with a(n-1) and has a prime factor not in a(n-2).

Original entry on oeis.org

1, 2, 4, 6, 3, 15, 5, 10, 8, 12, 9, 21, 7, 14, 16, 18, 20, 22, 11, 33, 24, 26, 13, 39, 27, 30, 25, 35, 28, 32, 34, 17, 51, 36, 38, 19, 57, 42, 40, 44, 46, 23, 69, 45, 48, 50, 52, 54, 56, 49, 63, 60, 55, 65, 70, 58, 29, 87, 66, 62, 31, 93, 72, 64, 68, 74, 37, 111, 75, 78, 76, 80, 82, 41, 123, 81
Offset: 1

Views

Author

Scott R. Shannon, Dec 12 2020

Keywords

Comments

Inspired by A064413 and A336957. The terms show a similar pattern to A064413, and like that sequence they are likely a permutation of the positive integers. Many terms also match the values in A169837. For example a(17)=20 to a(115)=111 (shifted by an index of 1) are the same, but then differ again before more matches occurr.
See A339670 for a similar sequence where the prime factor rules are reversed.

Examples

			a(4) = 6 as a(3) = 4 = 2*2 and a(2) = 2, thus a(4) must contain 2 as a prime factor but must also contain a prime factor other than 2. The lowest unused number matching these criteria is 2*3 = 6.
a(6) = 15 as a(5) = 3 and a(4) = 6 = 2*3, thus a(6) must contain 3 as a prime factor but must also contain a prime factor other than 2 and 3. The lowest unused number matching these criteria is 3*5 = 15. This is the first term that differs from A064413.

Crossrefs

Cf. A339670, A064413, A169837, A336957, A098550, A255582, A020639.

Programs

  • Mathematica
    Block[{a = {1, 2}, b = {}, c = {2}, p, k}, Do[k = 2; While[Nand[FreeQ[a, k], IntersectingQ[c, Set[p, FactorInteger[k][[All, 1]]]], Length@ Complement[p, Intersection[b, p]] > 0], k++]; AppendTo[a, k]; b = c; c = p, 75]; a] (* Michael De Vlieger, Dec 12 2020 *)

A375889 a(1) = 1, a(2) = 3; for n > 2, a(n) is the smallest unused positive number such that a(n) shares a factor with a(n-1) and (a(n) AND a(n-1)) = min(a(n), a(n-1)), where AND is the binary AND operation.

Original entry on oeis.org

1, 3, 15, 5, 45, 9, 27, 18, 2, 6, 4, 12, 8, 10, 14, 30, 16, 20, 22, 54, 32, 34, 38, 36, 39, 33, 51, 17, 85, 65, 75, 66, 64, 68, 70, 78, 72, 74, 90, 24, 26, 58, 40, 42, 46, 44, 60, 28, 62, 48, 50, 55, 35, 63, 7, 119, 21, 87, 69, 93, 81, 117, 52, 116, 80, 82, 86, 84, 92, 76, 94, 88, 120, 56, 122, 96
Offset: 1

Views

Author

Scott R. Shannon, Sep 01 2024

Keywords

Comments

The fixed points begin 1, 12, 284, 50726, 50764, 50770, 50772, 50811, although there are likely more.
The sequence is conjectured to be a permutation of the positive integers.

Examples

			a(4) = 5 as 5 shares a factor with a(3) = 15 and min(15,5) = 5 and (5 AND 15) = 101_2 AND 1111_2 = 101_2 = 5.

Links

Crossrefs

Cf. A375890, A303767, A169837, A027750.

A375890 a(1) = 1; for n > 1, a(n) is the smallest unused positive number such that a(n) is coprime to a(n-1) and (a(n) AND a(n-1)) = min(a(n), a(n-1)), where AND is the binary AND operation.

Original entry on oeis.org

1, 3, 2, 7, 4, 5, 13, 8, 9, 11, 10, 27, 16, 17, 19, 18, 23, 6, 31, 12, 29, 20, 21, 53, 32, 33, 35, 34, 39, 37, 36, 47, 14, 15, 79, 64, 65, 67, 66, 71, 68, 69, 77, 72, 73, 75, 74, 91, 24, 25, 57, 40, 41, 43, 42, 59, 26, 63, 22, 87, 70, 103, 38, 55, 48, 49, 51, 50, 119, 52, 61, 28, 93, 76, 109, 44
Offset: 1

Views

Author

Scott R. Shannon, Sep 01 2024

Keywords

Comments

The fixed points begin 1, 8, 9, 7453, 9338, although there are likely more.
The sequence is conjectured to be a permutation of the positive integers.

Examples

			a(5) = 4 as 4 is coprime to a(4) = 7 and min(7,4) = 4 and (7 AND 4) = 111_2 AND 100_2 = 100_2 = 4.

Links

Crossrefs

Cf. A375889, A303767, A169837, A027750.

A255198 Let EKG-n denote the EKG sequence (A064413) started with n rather than 2, and suppose EKG-n first merges with some other EKG-i (i >= 2) sequence after f(n) (= A255583(n)) steps; then a(n) = number of i such that EKG-i meets EKG-n after f(n) steps.

Original entry on oeis.org

1, 1, 1, 4, 1, 6, 2, 2, 5
Offset: 2

Views

Author

Gordon Hamilton, Feb 16 2015

Keywords

Comments

This sequence can be used in a classroom to introduce students to divisors.
For an explanatory video, see the Youtube link.
EKG-5 merges with EKG-2 after three steps, so some care is needed in the definition. Perhaps the offset should be 3 rather than 2? - N. J. A. Sloane, Feb 24 2015
Merging means that the sequences are identical for all future steps. EKG-2 and EKG-5 merge at step 44. From then on the sequences are identical.
EKG-3 and EKG-5 (below) do not merge at step 3, because the sequences are not identical from that point forward.

Examples

			a(5) = 4 because the EKG sequence starting with 5 (EKG-5, A169841) starts coinciding with sequences EKG-3, EKG-6, EKG-9 and EKG-12 simultaneously (when all sequences hit 18).
EKG-3:  3, 6, 2, 4, 8, 10, 5, 15, 9, 12, 14, 7, 21, 18, 16, 20, 22, 11, ... (A169837)
EKG-6:  6, 2, 4, 8, 10, 5, 15, 3, 9, 12, 14, 7, 21, 18, 16, 20, 22, 11, ... (A169843)
EKG-9:  9, 3, 6, 2, 4, 8, 10, 5, 15, 12, 14, 7, 21, 18, 16, 20, 22, 11, ... (A169849)
EKG-12: 12, 2, 4, 6, 3, 9, 15, 5, 10, 8, 14, 7, 21, 18, 16, 20, 22, 11, ... (A169855)
EKG-5:  5, 10, 2, 4, 6, 3, 9, 12, 8, 14, 7, 21, 15, 18, 16, 20, 22, 11, ... (A169841)
a(12) = 3 because the EKG sequence starting with 12 (EKG-12, A169855) starts coinciding with sequences EKG-3, EKG-6, and EKG-9 simultaneously (when all sequences hit 14).

Links

Crossrefs

Cf. A064413, A169837, A169841, A169843, A169849, A169857, A255583.
A255524 gives the smallest closest neighbor.
Previous Showing 11-18 of 18 results.

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

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