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A252865 a(n) = n if n <= 3, otherwise the smallest squarefree number not occurring earlier having at least one common factor with a(n-2), but none with a(n-1). 4

%I #23 Sep 02 2018 16:40:13

%S 1,2,3,10,21,5,6,35,22,7,11,14,33,26,15,13,30,91,34,39,17,42,85,38,51,

%T 19,66,95,46,55,23,65,69,70,57,58,93,29,31,87,62,105,74,77,37,110,111,

%U 82,129,41,43,123,86,141,106,47,53,94,159,118,165,59,78,295

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

%C Similar to A098550, but the restriction to squarefree makes it more a sequence of sets of primes, represented by their product.

%C The sequence has consecutive primes at indices 2 (2 & 3), 10 (7 & 11), 38 (29 & 31), 50 (41 & 43), and 56 (47 & 53). We conjecture that there are no further such pairs.

%H Reinhard Zumkeller, <a href="/A252865/b252865.txt">Table of n, a(n) for n = 1..10000</a>

%H David L. Applegate, Hans Havermann, Bob Selcoe, Vladimir Shevelev, N. J. A. Sloane, and Reinhard Zumkeller, <a href="http://arxiv.org/abs/1501.01669">The Yellowstone Permutation</a>, arXiv preprint arXiv:1501.01669, 2015 and <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Sloane/sloane9.html">J. Int. Seq. 18 (2015) 15.6.7</a>.

%t a[n_ /; n <= 3] = n;

%t a[n_] := a[n] = For[k = 1, True, k++, If[SquareFreeQ[k], If[FreeQ[Array[a, n-1], k], If[!CoprimeQ[k, a[n-2]] && CoprimeQ[k, a[n-1]], Return[k]]]]];

%t Array[a, 100] (* _Jean-François Alcover_, Sep 02 2018 *)

%o (PARI) invecn(v, k, x)=for(i=1, k, if(v[i]==x, return(i))); 0

%o alist(n)=local(v=vector(n,i,i), x); for(k=4, n, x=4; while(!issquarefree(x)||invecn(v, k-1, x)||gcd(v[k-2], x)==1||gcd(v[k-1],x)!=1, x++); v[k]=x); v

%o (Haskell)

%o import Data.List (delete)

%o a252865 n = a252865_list !! (n-1)

%o a252865_list = 1 : 2 : 3 : f 2 3 (drop 3 a005117_list) where

%o f u v ws = g ws where

%o g (x:xs) = if gcd x u > 1 && gcd x v == 1

%o then x : f v x (delete x ws) else g xs

%o -- _Reinhard Zumkeller_, Dec 24 2014

%o (Python)

%o from fractions import gcd

%o from sympy import factorint

%o A252865_list, l1, l2, s, b = [1,2,3], 3, 2, 4, set()

%o for _ in range(10**2):

%o ....i = s

%o ....while True:

%o ........if max(factorint(i).values()) == 1:

%o ............if not i in b and gcd(i,l1) == 1 and gcd(i,l2) > 1:

%o ................A252865_list.append(i)

%o ................l2, l1 = l1, i

%o ................b.add(i)

%o ................while s in b:

%o ....................b.remove(s)

%o ....................s += 1

%o ................break

%o ........else:

%o ............b.add(i)

%o ........i += 1 # _Chai Wah Wu_, Dec 24 2014

%Y Cf. A098550, A005117, A252867, A252868.

%Y Cf. A251391.

%K nonn

%O 1,2

%A _Franklin T. Adams-Watters_, Dec 23 2014

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Last modified March 28 09:04 EDT 2024. Contains 371240 sequences. (Running on oeis4.)