A251538 -id:A251538 - cp's OEIS frontend

A251539 First differences of A251538.

4, 4, 4, 4, 4, 4, 4, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 1, 1, 1, 1, 2, 4, 4, 4, 4, 4, 4, 4, 4, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 3, 4, 4

Offset: 1

N. J. A. Sloane, Dec 07 2014

nonn

I would very much like to have a formula or recurrence for this sequence.

Reinhard Zumkeller, Table of n, a(n) for n = 1..100000

Cf. A098548, A251535-A251538.

Cf. A251767 (duplicates removed), A251768 (run lengths).

a251539 n = a251539_list !! (n-1)
a251539_list = zipWith (-) (tail a251538_list) a251538_list
-- Reinhard Zumkeller, Dec 08 2014

max = 1000 (* = max term of A251538 *);
A098548 = {1, 2, 3};
For[n = 4, n <= 8 max, n++, If[GCD[n, A098548[[-1]]] == 1 && GCD[n, A098548[[-2]]] > 1, AppendTo[A098548, n]]];
A251538 = Select[Range[max], A098548[[2#+3]] > A098548[[2#+1]] + 6&];
Differences[A251538] (* Jean-François Alcover, Aug 01 2018 *)

A098548 a(n) = n if n <= 3, otherwise the smallest number > a(n-1) having at least one common factor with a(n-2) but none with a(n-1).

Original entry on oeis.org

1, 2, 3, 4, 9, 10, 21, 22, 27, 28, 33, 34, 39, 40, 51, 52, 57, 58, 63, 64, 69, 70, 81, 82, 87, 88, 93, 94, 99, 100, 111, 112, 117, 118, 123, 124, 129, 130, 141, 142, 147, 148, 153, 154, 159, 160, 171, 172, 177, 178, 183, 184, 189, 190, 201, 202, 207, 208, 213, 214

Offset: 1

Reinhard Zumkeller, Sep 14 2004

nonn

The number a(n) is even if and only if n is even. If n>=1, then a(2n) = a(2n-1) + 1. If n>=2, then a(2n+1) - a(2n) >= 5. As a consequence, if n>=15, then a(n) > 3n. - Benoit Jubin, Dec 07 2014

A098549(n) = a(a(n)).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
David L. Applegate, Hans Havermann, Bob Selcoe, Vladimir Shevelev, N. J. A. Sloane, and Reinhard Zumkeller, The Yellowstone Permutation, arXiv preprint arXiv:1501.01669 [math.NT], 2015 and J. Int. Seq. 18 (2015) 15.6.7.
Reinhard Zumkeller, Table of n, a(n) for n = 1..100000

Cf. A098549, A098550.
Cf. A158478 (smallest prime factor), A251104 (largest prime factor), A251139 (number of distinct prime factors), A251141 (total number of prime factors), A251046 (squarefree part), A251090 (squarefree kernel).
Cf. also A251535 and A251536 (bisections), A251537, A251538, A251539 (jumps), A251540.

a098548 n = a098548_list !! (n-1)
a098548_list = 1 : 2 : 3 : f 2 3 [4..] where
   f u v (w:ws) = if gcd u w > 1 && gcd v w == 1
                     then w : f v w ws else f u v ws
-- Reinhard Zumkeller, Nov 21 2014

x2 := 0: for n from 1 to 1000 do x := x2 + 1: while (n >= 4 and (gcd(x,x2) > 1 or gcd(x,x1) = 1)) do x := x + 1: end do; print (n, x); x1 := x2: x2 := x: end do: # David Applegate, Nov 26 2014

a := {1, 2, 3}; For[n = 4, n <= 1000, n++, If[GCD[n, a[[-1]]] == 1 && GCD[n, a[[-2]]] > 1, AppendTo[a, n]]]; a (* L. Edson Jeffery, Dec 04 2014 *)

A251537 Odd numbers n such that A098548(n+2) > A098548(n) + 6.

Original entry on oeis.org

5, 13, 21, 29, 37, 45, 53, 61, 67, 75, 83, 91, 99, 107, 115, 123, 131, 139, 147, 153, 161, 169, 177, 185, 193, 201, 209, 217, 225, 233, 239, 247, 255, 263, 271, 279, 287, 295, 303, 311, 319, 325, 333, 341, 349, 357, 365, 373

Offset: 1

N. J. A. Sloane, Dec 07 2014

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..100000

Cf. A098548, A251538, A251539.

a251537 n = a251537_list !! (n-1)
a251537_list = filter (\x -> a098548 (x + 2) > a098548 x + 6) [1, 3 ..]
-- Reinhard Zumkeller, Dec 08 2014

A251767 Consecutive repeated terms removed from in A251539.

Original entry on oeis.org

4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 1, 2, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 1, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 1, 4, 3, 4, 3, 4, 2, 4, 3, 4, 3, 4, 3, 4, 1, 2, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 1, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 2, 4, 1, 4, 3

Offset: 1

Reinhard Zumkeller, Dec 08 2014

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A251539, A251768.

import Data.List (group)
a251767 n = a251767_list !! (n-1)
a251767_list = map head $ group a251539_list

max = 2000 (* = max term of A251538 *);
A098548 = {1, 2, 3};
For[n = 4, n <= 8 max, n++, If[GCD[n, A098548[[-1]]] == 1 && GCD[n, A098548[[-2]]] > 1, AppendTo[A098548, n]]];
A251538 = Select[Range[max], A098548[[2 # + 3]] > A098548[[2 # + 1]] + 6&];
A251539 = Differences[A251538];
First /@ Split[A251539] (* Jean-François Alcover, Aug 01 2018 *)

A251768 Run lengths in A251539.

Original entry on oeis.org

7, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 4, 1, 8, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 2, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 2, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 3, 1, 8, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 2, 10, 1, 10, 1, 10

Offset: 1

Reinhard Zumkeller, Dec 08 2014

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A249943, A251767.

import Data.List (group)
a251768 n = a251768_list !! (n-1)
a251768_list = map length $ group a251539_list

max = 2000 (* = max term of A251538 *);
A098548 = {1, 2, 3};
For[n = 4, n <= 8 max, n++, If[GCD[n, A098548[[-1]]] == 1 && GCD[n, A098548[[-2]]] > 1, AppendTo[A098548, n]]];
A251538 = Select[Range[max], A098548[[2 # + 3]] > A098548[[2 # + 1]] + 6&];
A251539 = Differences[A251538];
Length /@ Split[A251539] (* Jean-François Alcover, Aug 01 2018 *)

cp's OEIS Frontend

A251539 First differences of A251538.

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A098548 a(n) = n if n <= 3, otherwise the smallest number > a(n-1) having at least one common factor with a(n-2) but none with a(n-1).

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A251537 Odd numbers n such that A098548(n+2) > A098548(n) + 6.

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Haskell

A251767 Consecutive repeated terms removed from in A251539.

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Haskell

Mathematica

A251768 Run lengths in A251539.

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Mathematica