A098550 -id:A098550 - cp's OEIS frontend

A251239 Indices of prime numbers in A098550.

2, 3, 9, 15, 22, 23, 30, 43, 51, 61, 62, 79, 87, 88, 101, 114, 127, 132, 142, 153, 158, 167, 175, 194, 204, 215, 222, 233, 238, 247, 274, 283, 296, 301, 324, 329, 338, 355, 364, 375, 386, 393, 414, 423, 430, 435, 452, 479, 490, 497, 506, 523, 528, 541, 550

Offset: 1

Reinhard Zumkeller, Dec 02 2014

nonn

It is conjectured that every prime appears in A098550, and if so then A098550(a(n)) = A000040(n). [Comment edited by N. J. A. Sloane, Dec 15 2014] [It is now known that every prime appears in A098550, although it is not known that they appear in their right order. - N. J. A. Sloane, Dec 25 2014]
A010051(A098550(a(n))) = 1; A049084(A098550(a(n))) > 0.
Conjecture: a(n) = A251541(n) + 2 for n > 4. - Reinhard Zumkeller, Dec 16 2014
A253049(n) = A098550(a(n)+1). - Reinhard Zumkeller, Dec 29 2014

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

Cf. A098550, A010051, A049084, A251392, A247253 (first differences), A251595, A251541, A253049, A253048.

This is a subsequence of A251391 and A251241,

a251239 n = a251239_list !! (n-1)
a251239_list = filter ((== 1) . a010051' . a098550) [1..]

a098550[lst_List] :=
Block[{k = 4},
  While[GCD[lst[[-2]], k] == 1 || GCD[lst[[-1]], k] > 1 ||
    MemberQ[lst, k], k++]; Append[lst, k]];
a251239[n_] :=
Flatten@Position[Nest[a098550, {1, 2, 3}, n], Integer?PrimeQ]; a251239[550] (* _Michael De Vlieger, Dec 23 2014, based on Robert G. Wilson v at A098550 *)

A098551 Inverse of A098550.

Original entry on oeis.org

1, 2, 3, 4, 9, 10, 15, 6, 5, 16, 22, 12, 23, 8, 7, 14, 30, 31, 43, 18, 17, 20, 51, 33, 11, 25, 19, 27, 61, 39, 62, 29, 24, 35, 13, 37, 79, 41, 21, 48, 87, 44, 88, 46, 26, 56, 101, 52, 40, 50, 28, 54, 114, 69, 34, 58, 47, 63, 127, 71, 132, 60, 42, 65, 36, 73, 142, 67, 49, 80, 153

Offset: 1

Reinhard Zumkeller, Sep 14 2004

nonn

Now known to be a permutation of the natural numbers: see the 2015 article by Applegate, Havermann, Selcoe, Shevelev, Sloane, and Zumkeller.

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.
Index entries for sequences that are permutations of the natural numbers

Cf. A098550, A098553.
Cf. A249943 (partial maxima).
Cf. A098479, A255940.

import Data.List (elemIndex); import Data.Maybe (fromJust)
a098551 = (+ 1) . fromJust . (`elemIndex` a098550_list)
-- Reinhard Zumkeller, Nov 21 2014

f[lst_List] := Block[{k = 4}, While[ GCD[ lst[[-2]], k] == 1 || GCD[ lst[[-1]], k] > 1 || MemberQ[lst, k], k++]; Append[lst, k]]; Table[ Position[ Nest[ f, {1, 2, 3}, 120], n], {n, 71}] // Flatten (* Robert G. Wilson v, Nov 21 2014 *)

A098553(n) = a(a(n)).

A251416 a(n) = smallest number not in {A098550(1), A098550(2), ..., A098550(n)}.

Original entry on oeis.org

2, 3, 4, 5, 5, 5, 5, 5, 6, 7, 7, 7, 7, 7, 10, 11, 11, 11, 11, 11, 11, 13, 17, 17, 17, 17, 17, 17, 17, 18, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 23, 23, 23, 23, 23, 23, 23, 23, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 31

Offset: 1

N. J. A. Sloane, Dec 03 2014

nonn

a(n) = smallest missing number in A098550 once we have found A098550(n).

Bradley Klee conjectures that after a(30)=18, all further terms are primes, that every prime appears, and the primes appear in increasing order.

Bradley Klee, Posting to Sequence Fans Mailing List, Dec 03 2014

N. J. A. Sloane and Reinhard Zumkeller, Table of n, a(n) for n = 1..100000 (first 1000 terms from N. J. A. Sloane)

Cf. A098550, A251415. See A251417 for lengths of runs.
Cf. A251595 (distinct terms).
Cf. A251546, A251549.

import Data.List (delete)
a251416 n = a251416_list !! (n-1)
a251416_list = 2 : 3 : f 2 3 [4..] where
   f u v ws = h ws where
     h (x:xs) = if gcd x u > 1 && gcd x v == 1
                   then (head ws) : f v x (delete x ws) else h xs
-- Reinhard Zumkeller, Dec 05 2014

# This produces the first 100 terms. Uses b1 = list of terms in A098550, from b-file
b2:={$3..5000}:
b3:=[2]:
for i from 2 to 100 do
b2:=remove('x->x=b1[i]',b2):
b3:=[op(b3),b2[1]];
od:
b3;

terms = 100;
f[lst_List] := Block[{k = 4}, While[GCD[lst[[-2]], k] == 1 || GCD[lst[[-1]], k] > 1 || MemberQ[lst, k], k++]; Append[lst, k]];
A098550 = Nest[f, {1, 2, 3}, terms-3];
a[1] = 2;
a[n_] := a[n] = For[k = a[n - 1], True, k++, If[FreeQ[A098550[[1 ;; n]], k], Return[k]]];
Array[a, terms] (* Jean-François Alcover, Aug 01 2018, after Robert G. Wilson v *)

a(n) = Min{A251546(n), A251549(n)}. - Reinhard Zumkeller, Dec 19 2014

A251546 a(n) = smallest even number not in {A098550(1), A098550(2), ..., A098550(n)}.

Original entry on oeis.org

2, 4, 4, 6, 6, 6, 6, 6, 6, 10, 10, 10, 10, 10, 10, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 24, 24, 30, 30, 30, 30, 30, 30, 38, 38, 40, 40, 40, 40, 40, 40, 40, 46, 46, 46, 46, 46, 46, 46, 46, 54, 54, 54

Offset: 1

N. J. A. Sloane, Dec 18 2014

nonn

A251416(n) = Min{a(n), A251549(n)}. - Reinhard Zumkeller, Dec 19 2014

Chai Wah Wu, 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.

Cf. A098550, A247253, A251416, A251417, A251547, A251548, A251549, A251550, A251551, A251552.

A251544 List of values of A098550(n+2) for those n for which A098550(n) is a prime.

Original entry on oeis.org

4, 9, 25, 21, 33, 26, 85, 95, 115, 58, 93, 111, 82, 129, 329, 265, 177, 427, 335, 213, 365, 237, 581, 267, 679, 505, 309, 321, 545, 565, 381, 655, 411, 973, 1043, 755, 785, 1141, 835, 519, 895, 1267, 1337, 579, 985, 597, 633, 1115, 1135, 687, 699, 1673, 723, 753

Offset: 1

David Applegate and N. J. A. Sloane, Dec 16 2014

nonn

For n > 4: third column in A251637. - Reinhard Zumkeller, Dec 16 2014

N. J. A. Sloane, Table of n, a(n) for n = 1..11246
David L. Applegate, Hans Havermann, Bob Selcoe, Vladimir Shevelev, N. J. A. Sloane, and Reinhard Zumkeller, The Yellowstone Permutation, arXiv preprint arXiv:1501.01669, 2015 and J. Int. Seq. 18 (2015) 15.6.7.

Cf. A098550, A251542 (another version). See A251545 for the sorted values.

Cf. A251637.

max = 600;
f[lst_] := Block[{k = 4}, While[GCD[lst[[-2]], k] == 1 || GCD[lst[[-1]], k] > 1 || MemberQ[lst, k], k++]; Append[lst, k]];
A098550 = Nest[f, {1, 2, 3}, max - 3];
sel = Select[Transpose[{Range[max], A098550}], PrimeQ[#[[2]]]&][[All,1]]+2;
A098550[[sel]] (* Jean-François Alcover, Sep 05 2018, after Robert G. Wilson v in A098550 *)

A251542 List of values A098550(n+2)/A098550(n) for those n for which A098550(n) is a prime.

Original entry on oeis.org

2, 3, 5, 3, 3, 2, 5, 5, 5, 2, 3, 3, 2, 3, 7, 5, 3, 7, 5, 3, 5, 3, 7, 3, 7, 5, 3, 3, 5, 5, 3, 5, 3, 7, 7, 5, 5, 7, 5, 3, 5, 7, 7, 3, 5, 3, 3, 5, 5, 3, 3, 7, 3, 3, 5, 3, 5, 7, 3, 5, 7, 3, 3, 5, 11, 3, 5, 5, 5, 3, 5, 5, 5, 5, 7, 3, 7, 5, 5, 7, 3, 5, 5, 3, 3, 5, 3, 7, 7, 5

Offset: 1

David Applegate and N. J. A. Sloane, Dec 15 2014

nonn

a(n) <= 17 for n <= 250000 (see A251543).
For n > 4: third column in A251715. - Reinhard Zumkeller, Dec 16 2014
a(n) <= 19 for n <= 10^6. - Chai Wah Wu, Dec 16 2014

			A098550(n) for n= 1..11 is 1,2,3,4,9,8,15,14,5,6,25. Each time you see a prime, divide the term two steps ahead by that prime. The result is 4/2=2, 9/3=3, 25/5=5,...

Chai Wah Wu, 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, 2015 and J. Int. Seq. 18 (2015) 15.6.7.

Cf. A098550, A251543. See A251544 for the actual values of A098550(n+2).

Cf. A251715.

a251542 n = a251542_list !! (n-1)
a251542_list = [div u v | (u, v) <- zip(drop 2 a098550_list) a098550_list,
                          a010051' v == 1]
-- Reinhard Zumkeller, Mar 11 2015

max = 1200;
f[lst_] := Block[{k = 4}, While[GCD[lst[[-2]], k] == 1 || GCD[lst[[-1]], k] > 1 || MemberQ[lst, k], k++]; Append[lst, k]];
A098550 = Nest[f, {1, 2, 3}, max - 3];
sel = Select[Transpose[{Range[max], A098550}], PrimeQ[#[[2]]]&][[All,1]]+2;
A098550[[sel]]/A098550[[sel - 2]] (* Jean-François Alcover, Sep 05 2018, after Robert G. Wilson v in A098550 *)

A251604 A Zumkeller-type sequence (cf. A098550): a(n) = n if n <= 3, otherwise the smallest number not occurring earlier having at least one common factor with a(n-2)+a(n-1), but none with a(n-1).

Original entry on oeis.org

1, 2, 3, 5, 4, 9, 13, 6, 19, 10, 29, 12, 41, 53, 8, 61, 15, 14, 87, 101, 16, 21, 37, 18, 11, 58, 23, 24, 47, 71, 20, 7, 27, 17, 22, 39, 122, 35, 157, 26, 33, 59, 28, 45, 73, 30, 103, 38, 51, 89, 25, 32, 57, 178, 55, 233, 34, 63, 97, 36, 49, 40, 267, 307, 42

Offset: 1

Vladimir Shevelev, Dec 13 2014

nonn

Conjectured to be a permutation of the positive integers.

See also A255972 for this conjecture. - Reinhard Zumkeller, Mar 12 2015

Peter J. C. Moses, Table of n, a(n) for n = 1..5000
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.

Cf. A098550.

Cf. A255972.

import Data.List (delete)
a251604 n = a251604_list !! (n-1)
a251604_list = 1 : 2 : 3 : f 2 3 [4..] where
   f u v ws = g ws where
     g (x:xs) = if gcd x (u + v) > 1 && gcd x v == 1
                   then x : f v x (delete x ws) else g xs
-- Reinhard Zumkeller, Mar 12 2015

a[n_] := a[n] = If[n <= 3, n, For[k = 1, True, k++, If[FreeQ[Array[a, n-1], k], If[!CoprimeQ[k, a[n-2]+a[n-1]] && CoprimeQ[k, a[n-1]], Return[k]]]]];
Array[a, 65] (* Jean-François Alcover, Jul 31 2018 *)

More terms from Peter J. C. Moses, Dec 13 2014

A249943 a(n) = smallest k such that the numbers 1..n appear among A098550(1), ..., A098550(k), or a(n) = 0 if there is no such k.

Original entry on oeis.org

1, 2, 3, 4, 9, 10, 15, 15, 15, 16, 22, 22, 23, 23, 23, 23, 30, 31, 43, 43, 43, 43, 51, 51, 51, 51, 51, 51, 61, 61, 62, 62, 62, 62, 62, 62, 79, 79, 79, 79, 87, 87, 88, 88, 88, 88, 101

Offset: 1

Vladimir Shevelev, Dec 04 2014

nonn

The conjecture that all terms are positive is equivalent to the known conjecture that A098550 is a permutation of the positive integers.

Partial maxima of A098551: a(n) = max{a(n-1),A098551(n)} for n > 1. - Reinhard Zumkeller, Dec 06 2014

			Let n=6. Since A098550(9)=5 and A098550(10)=6, a(6)=10. - Corrected by _David Applegate_, Dec 08 2014

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, 2015 and J. Int. Seq. 18 (2015) 15.6.7.

Cf. A098550, A098551.

Cf. A251620 (duplicates removed), A251621 (run lengths).

a249943 n = a249943_list !! (n-1)
a249943_list = scanl1 max $ map a098551 [1..]
-- Reinhard Zumkeller, Dec 06 2014

f[lst_List] := Block[{k=4}, While[GCD[lst[[-2]], k] == 1 || GCD[lst[[-1]], k]>1 || MemberQ[lst, k], k++]; Append[lst, k]]; A098550 = Nest[f, {1, 2, 3}, 100]; runningMax := Rest[FoldList[Max, -Infinity, #]]&; runningMax[Take[Ordering[A098550], NestWhile[#+1&, 1, MemberQ[A098550, #]&]-1]] (* Jean-François Alcover, Dec 05 2014, after Robert G. Wilson v and Peter J. C. Moses *)

The author conjectures that a(n)/n <= a(19)/19 = 43/19. Peter J. C. Moses verified that the strict inequality holds for 19 < n <= 1.1*10^5. - Vladimir Shevelev, Dec 06 2014

A251237 Indices of even numbers in A098550.

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 25, 27, 29, 31, 33, 35, 37, 39, 41, 44, 46, 48, 50, 52, 54, 56, 58, 60, 63, 65, 67, 69, 71, 73, 75, 77, 80, 82, 84, 86, 89, 91, 93, 95, 97, 99, 102, 104, 106, 108, 110, 112, 115, 117, 119, 121, 123, 125, 128, 130, 133, 135

Offset: 1

Reinhard Zumkeller, Dec 02 2014

nonn

A098550(a(n)) mod 2 = 0;
a(n+1) > a(n) + 1;
there are infinitely many even terms in A098550, for proof: see comment #4 in A098550.

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, 2015 and J. Int. Seq. 18 (2015) 15.6.7.

Cf. A098550, A005843, A251238, A251393.

First row of array A251716.

a251237 n = a251237_list !! (n-1)
a251237_list = filter (even . a098550) [1..]

f[lst_] := Block[{k = 4}, While[GCD[lst[[-2]], k] == 1 || GCD[lst[[-1]], k] > 1 || MemberQ[lst, k], k++]; Append[lst, k]];
Position[Nest[f, {1, 2, 3}, 140], ?EvenQ] // Flatten (* _Jean-François Alcover, Oct 01 2018, after Robert G. Wilson v in A098550 *)

A251241 Indices of prime powers in A098550.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 9, 11, 14, 15, 19, 22, 23, 29, 30, 40, 43, 51, 57, 61, 62, 65, 79, 87, 88, 94, 101, 114, 124, 127, 132, 137, 142, 153, 158, 167, 171, 175, 187, 194, 204, 215, 222, 233, 238, 247, 269, 273, 274, 277, 283, 296, 301, 313, 324, 329, 338, 355

Offset: 1

Reinhard Zumkeller, Dec 02 2014

nonn

A010055(A098550(a(n))) = 1; for n > 0: A100995(A098550(a(n))) > 0;

A098550(a(n)) = A000961(n).

Reinhard Zumkeller, Table of n, a(n) for n = 1..5000

Cf. A098550, A000961, A010055, A100995, subsequences: A251239 and A251240.

a251241 n = a251241_list !! (n-1)
a251241_list = filter ((== 1) . a010055 . a098550) [1..]

cp's OEIS Frontend

A251239 Indices of prime numbers in A098550.

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A098551 Inverse of A098550.

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A251416 a(n) = smallest number not in {A098550(1), A098550(2), ..., A098550(n)}.

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A251546 a(n) = smallest even number not in {A098550(1), A098550(2), ..., A098550(n)}.

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A251544 List of values of A098550(n+2) for those n for which A098550(n) is a prime.

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A251542 List of values A098550(n+2)/A098550(n) for those n for which A098550(n) is a prime.

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A251604 A Zumkeller-type sequence (cf. A098550): a(n) = n if n <= 3, otherwise the smallest number not occurring earlier having at least one common factor with a(n-2)+a(n-1), but none with a(n-1).

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A249943 a(n) = smallest k such that the numbers 1..n appear among A098550(1), ..., A098550(k), or a(n) = 0 if there is no such k.

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