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%I #21 Oct 03 2018 08:56:37
%S 1,2,3,10,21,5,6,35,22,15,14,33,7,30,77,26,105,13,42,65,66,91,11,70,
%T 143,210,187,39,55,78,385,34,165,182,51,110,273,85,154,195,119,330,17,
%U 231,170,429,238,715,102,455,374,1365,38,1155,442,57,770,663,95,462,1105,114,1001,255
%N Squarefree version of A252867.
%C a(n) = n if n <= 3, otherwise the first squarefree number not occurring earlier having at least one common factor with a(n-2), but none with a(n-1). The squarefree numbers are ordered by their occurrence in A019565.
%C These represent the same sets of integers as A252867 does, but using the factorization of squarefree numbers for the representation.
%C This is a permutation of the squarefree numbers. [I believe this is at present only a conjecture. - _N. J. A. Sloane_, Jan 10 2015]
%H Chai Wah Wu, <a href="/A252868/b252868.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>.
%F a(n)=A019565(A252867(n-1))
%t (* b = A019565, c = A252867 *)
%t b[n_] := Times @@ Prime[Flatten[Position[#, 1]]]&[Reverse[IntegerDigits[n, 2]]];
%t c[n_] := c[n] = If[n<3, n, For[k=3, True, k++, If[FreeQ[Array[c, n-1], k], If[BitAnd[k, c[n-2]] >= 1 && BitAnd[k, c[n-1]] == 0, Return[k]]]]];
%t a[n_] := b[c[n-1]];
%t Array[a, 64] (* _Jean-François Alcover_, Oct 03 2018 *)
%o (PARI) invecn(v, k, x)=for(i=1, k, if(v[i]==x, return(i))); 0
%o squarefree(n)=local(r=1,i=1);while(n>0,if(n%2,r*=prime(i));i++;n\=2);r
%o alist(n)=local(v=vector(n, i, i-1), x); for(k=4, n, x=3; while(invecn(v, k-1, x)||!bitand(v[k-2], x)||bitand(v[k-1], x), x++); v[k]=x); vector(n,i,squarefree(v[i]))
%o (Python)
%o from operator import mul
%o from functools import reduce
%o from sympy import prime
%o def A019565(n):
%o ....return reduce(mul,(prime(i+1) for i,v in enumerate(bin(n)[:1:-1]) if v == '1')) if n > 0 else 1
%o A252868_list, l1, l2, s, b = [1,2,3], 2, 1, 3, set()
%o for _ in range(10**4):
%o ....i = s
%o ....while True:
%o ........if not (i in b or i & l1) and i & l2:
%o ............A252868_list.append(A019565(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 ........i += 1 # _Chai Wah Wu_, Dec 25 2014
%Y Cf. A252867, A098550, A252865, A048793, A019565.
%K nonn
%O 1,2
%A _Franklin T. Adams-Watters_, Dec 23 2014
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