cp's OEIS Frontend

A168521 Sort numbers by value of sum of squares of prime factors (cf. A067666). Break ties by putting smaller numbers first. Begin with 0, 1.

%I A168521 #16 Feb 07 2025 00:51:13
%S A168521 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 A168521 192,15,144,108,81,512,40,384,30,288,216,162,1024,80,768,60,576,45,
%U A168521 432,324,2048,160,243,1536,120,1152,90,864,648,4096,7,320,486,3072,25,240
%N A168521 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 A168521 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 A168521 Keith Flower, <a href="http://quexotic.org/?p=46">Integers in Euclidean space order</a>
%H A168521 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A168521 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 A168521 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 A168521 Explanatory table for initial terms:
%e A168521   n  a(n)   P_{a(n)}
%e A168521   1    0                (appears here as prescribed)
%e A168521   2    1                (appears here as prescribed)
%e A168521                       Calculation of d^2
%e A168521   3    2 -> [2]       -> 2^2 = 4
%e A168521   4    4 -> [2,2]     -> 2^2 + 2^2 = 8
%e A168521   5    3 -> [3]       -> 3^3 = 9
%e A168521   6    8 -> [2,2,2]   -> 2^2 + 2^2 + 2^2 = 12
%e A168521   7    6 -> [2,3]     -> 2^2 + 3^2 = 13
%e A168521   8   16 -> [2,2,2,2] -> 2^2 + 2^2 + 2^2 + 2^2 = 16
%e A168521   9   12 -> [2,2,3]   -> 2^2 + 2^2 + 3^2 = 17
%Y A168521 Cf. A001222, A027746, A067666.
%Y A168521 Similarly defined sequences: A064364, A178595.
%K A168521 nonn,easy
%O A168521 1,3
%A A168521 Keith Flower (keith.flower(AT)gmail.com), Nov 28 2009
%E A168521 Definition edited by _N. J. A. Sloane_, Nov 29 2009
%E A168521 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 A168521 More terms from _R. J. Mathar_, Jan 25 2010
%E A168521 Edited by _Peter Munn_, Aug 17 2022