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%I #13 Aug 17 2022 22:31:44
%S 0,1,2,4,3,8,6,16,12,9,32,24,18,64,5,48,36,27,128,10,96,72,54,256,20,
%T 192,15,144,108,81,512,40,384,30,288,216,162,1024,80,768,60,576,45,
%U 432,324,2048,160,243,1536,120,1152,90,864,648,4096,7,320,486,3072,25,240
%N Sort numbers by value of sum of squares of prime factors (cf. A067666). Break ties by putting smaller numbers first. Begin with 0, 1.
%C Represent each number m by a corresponding point, P_m, in Euclidean space, such that the prime factors of m are the co-ordinates of P_m. In this sequence, the numbers appear in order of distance from the origin of their corresponding points.
%H Keith Flower, <a href="http://quexotic.org/?p=46">Integers in Euclidean space order</a>
%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F For n >= 2, Sum_{k=1..A001222(a(n))} A027746(a(n),k)^2 <= Sum_{k=1..A001222(a(n+1))} A027746(a(n+1),k))^2. - _Peter Munn_, Aug 17 2022
%e For m = 7, distance d from the origin of P_7 is 7, for m = 8192 (P_8192 = [2,2,2,2,2,2,2,2,2,2,2,2,2]) d = sqrt(13*2^2) = 7.211102550927978. So 7 appears before 8192.
%e Explanatory table for initial terms:
%e n a(n) P_{a(n)}
%e 1 0 (appears here as prescribed)
%e 2 1 (appears here as prescribed)
%e Calculation of d^2
%e 3 2 -> [2] -> 2^2 = 4
%e 4 4 -> [2,2] -> 2^2 + 2^2 = 8
%e 5 3 -> [3] -> 3^3 = 9
%e 6 8 -> [2,2,2] -> 2^2 + 2^2 + 2^2 = 12
%e 7 6 -> [2,3] -> 2^2 + 3^2 = 13
%e 8 16 -> [2,2,2,2] -> 2^2 + 2^2 + 2^2 + 2^2 = 16
%e 9 12 -> [2,2,3] -> 2^2 + 2^2 + 3^2 = 17
%Y Cf. A001222, A027746, A067666.
%Y Similarly defined sequences: A064364, A178595.
%K nonn,easy
%O 1,3
%A Keith Flower (keith.flower(AT)gmail.com), Nov 28 2009
%E Definition edited by _N. J. A. Sloane_, Nov 29 2009
%E It would also be worthwhile computing the companion sequence where ties are broken according to lexicographic order of the lists of prime factors (so that 48 would come before 5, instead of after). - _N. J. A. Sloane_, Nov 29 2009
%E More terms from _R. J. Mathar_, Jan 25 2010
%E Edited by _Peter Munn_, Aug 17 2022