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%I A302781 #54 Mar 09 2024 01:48:47
%S A302781 1,2,6,3,15,5,10,30,120,40,20,60,12,24,8,4,28,84,168,56,14,7,21,42,
%T A302781 210,105,35,70,280,840,420,140,1260,3780,7560,2520,630,315,945,1890,
%U A302781 378,189,63,126,504,1512,756,252,36,72,216,108,540,180,360,1080,270,90,45,135,27,54,18,9,117,351,702,234,936,468
%N A302781 Divisor-or-multiple permutation of natural numbers constructed from two-dimensional Hilbert curve (A163357) and Fermi-Dirac primes (A050376).
%C A302781 Note that the starting offset is 0, to align with A052330 and A207901.
%C A302781 Shares with A064736, A207901, A298480, A302350, A302783, A303771, etc. the property that a(n) is either a divisor or a multiple of a(n+1). Permutations satisfying such property are called "divisor-or-multiple permutations" in the OEIS, although Mazet & Saias call them "chain permutations" in their paper. [Edited by _Antti Karttunen_, Aug 26 2018]
%C A302781 One way to construct such permutations is by composing A052330 from the right with any such permutation like A003188 or A302846 where the binary expansions of a(n) and a(n+1) always differ by just a single bit-position.
%C A302781 Further permutations satisfying the same condition could be constructed from higher-dimensional versions (i.e., greater than 2) of Hilbert's space-filling curves, where the coordinates of each dimension would be Gray coded separately and then interleaved together. Permutation A207901 is essentially a one-dimensional variant of the same idea, while this is constructed from the 2-dimensional curve A163357, which is a Hamiltonian path on N X N grid.
%C A302781 See _Peter Munn_'s A300012 for another idea for constructing such a permutation. - _Antti Karttunen_, Aug 26 2018
%H A302781 Antti Karttunen, <a href="/A302781/b302781.txt">Table of n, a(n) for n = 0..16383</a>
%H A302781 Pierre Mazet and Eric Saias, <a href="https://arxiv.org/abs/1803.10073">Etude du graphe divisoriel 4</a>, arXiv:1803.10073 [math.NT], 2018.
%H A302781 Various, <a href="http://list.seqfan.eu/oldermail/seqfan/2018-April/018623.html">Discussion on SeqFan-list</a>, April 2018.
%H A302781 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A302781 a(n) = A052330(A302846(n)), where A302846(n) = A000695(A003188(A059253(n))) + 2*A000695(A003188(A059252(n))).
%F A302781 a(n) = A207901(A302844(n)) = A052330(A064706(A163356(n))).
%o A302781 (PARI)
%o A302781 up_to_e = 14;
%o A302781 v050376 = vector(up_to_e);
%o A302781 A050376(n) = v050376[n];
%o A302781 ispow2(n) = (n && !bitand(n,n-1));
%o A302781 i = 0; for(n=1,oo,if(ispow2(isprimepower(n)), i++; v050376[i] = n); if(i == up_to_e,break));
%o A302781 A052330(n) = { my(p=1,i=1); while(n>0, if(n%2, p *= A050376(i)); i++; n >>= 1); (p); };
%o A302781 A064706(n) = bitxor(n, n>>2);
%o A302781 A057300(n) = { my(t=1,s=0); while(n>0, if(1==(n%4),n++,if(2==(n%4),n--)); s += (n%4)*t; n >>= 2; t <<= 2); (s); };
%o A302781 A163356(n) = if(!n,n,my(i = (#binary(n)-1)\2, f = 4^i, d = (n\f)%4, r = (n%f)); (((((2+(i%2))^d)%5)-1)*f) + if(3==d,f-1-A163356(r),A057300(A163356(r))));
%o A302781 A302781(n) = A052330(A064706(A163356(n)));
%Y A302781 Cf. A302782 (inverse).
%Y A302781 Cf. A003188, A050376, A052330, A059252, A059253, A064706, A163356, A163357, A207901, A298480, A300012, A302844, A302846, A302783, A303771.
%K A302781 nonn
%O A302781 0,2
%A A302781 _Antti Karttunen_, Apr 14 2018
%E A302781 Name edited by _Antti Karttunen_, Aug 26 2018

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