A247225 a(n) = n if n <= 3, a(4)=5, otherwise the smallest number not occurring earlier having at least one common factor with a(n-3), but none with a(n-1)*a(n-2).

1, 2, 3, 5, 4, 9, 25, 8, 21, 55, 16, 7, 11, 6, 35, 121, 12, 49, 143, 10, 63, 13, 20, 27, 91, 22, 15, 119, 26, 33, 17, 14, 39, 85, 28, 57, 65, 32, 19, 45, 34, 133, 69, 40, 77, 23, 18, 175, 253, 24, 95, 161, 36, 125, 203, 38, 75, 29, 44, 51, 145, 46, 81, 155, 52

Offset: 1

Author

Vladimir Shevelev, Jan 11 2015

Keywords

• nonn

Comments

Conjecturally the sequence is a permutation of the positive integers. However, to prove this we need more subtle arguments than were used to prove the corresponding property for A098550. - Vladimir Shevelev, Jan 14 2015

For n <= 2000, a(3n-1) is even and both a(3n) and a(3n-2) are odd numbers. I conjecture that this is true for all positive integers n. This conjecture is true iff for all positive integers n, a(3n-1) is even. - Farideh Firoozbakht, Jan 14 2015

From Vladimir Shevelev, Jan 19 2015: (Start)

A generalization of A098550 and A247225.

Let p_n=prime(n). Define the following sequence

a(1)=1, a(2)=p_1,...,a(k+2)=p_(k+1), otherwise the smallest number not occurring earlier having at least one common factor with a(n-(k+1)), but none with a(n-1)*a(n-2)*...*a(n-k).

The sequence begins

1, p_1, p_2, ..., p_(k+1), p_1^2, p_2^2, ..., p_(k+1)^2, p_1^3, ... (*)

[ p_1^3 is followed by p_2*p_(k+2), k<=2,

p_2^3, k>=3, etc.]

In particular, if k=1, it is A098550, if k=2, it is A247225.

Conjecturally for every k>=2, as in the case k=1, the sequence (*) is a permutation of the positive integers. For k>=3, at first glance, already the appearance of the number 6 seems problematic. However, at the author's request, Peter J. C. Moses found that the positions of 6 are 83, 157, 1190, 206, ... in cases k=3,4,5,6,... respectively (A254003).

Note also that for every k>=2, every even term is followed by k odd terms. This is explained by the minimal growth of even numbers (2n) relatively with one of the numbers with the smallest prime divisor p>=3 (asymptotically 6n, 15n, 105n/4, 385n/8, ... for p = 3,5,7,11,... respectively (cf. A084967 - A084970)).

(End)

Links

• Peter J. C. Moses, Table of n, a(n) for n = 1..2000
• 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.

Crossrefs

Cf. A098550, A247942, A249167, A251604, A254003.

Programs

Mathematica

a[n_ /; n <= 3] := n; a[4]=5; a[n_] := a[n] = For[aa = Table[a[j], {j, 1, n-1}]; k=4, True, k++, If[FreeQ[aa, k] && !CoprimeQ[k, a[n-3]] && CoprimeQ[k, a[n-1]*a[n-2]], Return[k]]]; Table[ a[n], {n, 1, 65}] (* Jean-François Alcover, Jan 12 2015 *)

Extensions

More terms from Peter J. C. Moses, Jan 12 2015