

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(n3), but none with a(n1)*a(n2).


6



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
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OFFSET

1,2


COMMENTS

Conjecturally the sequence is a permutation of the natural numbers. 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(3n1) is even and both a(3n) and a(3n2) 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(3n1) 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(n1)*a(n2)*...*a(nk).
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 case k=1, the sequence (*) is a permutation of the natural numbers. For k>=3, on the first glance, already the appearance of the number 6 seems problematical. 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. It 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.


MATHEMATICA

a[n_ /; n <= 3] := n; a[4]=5; a[n_] := a[n] = For[aa = Table[a[j], {j, 1, n1}]; k=4, True, k++, If[FreeQ[aa, k] && !CoprimeQ[k, a[n3]] && CoprimeQ[k, a[n1]*a[n2]], Return[k]]]; Table[ a[n], {n, 1, 65}] (* JeanFrançois Alcover, Jan 12 2015 *)


CROSSREFS

Cf. A098550, A247942, A249167, A251604, A254003.
Sequence in context: A332301 A069202 A244984 * A100932 A064360 A075158
Adjacent sequences: A247222 A247223 A247224 * A247226 A247227 A247228


KEYWORD

nonn


AUTHOR

Vladimir Shevelev, Jan 11 2015


EXTENSIONS

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


STATUS

approved



