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.

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, 192, 15, 144, 108, 81, 512, 40, 384, 30, 288, 216, 162, 1024, 80, 768, 60, 576, 45, 432, 324, 2048, 160, 243, 1536, 120, 1152, 90, 864, 648, 4096, 7, 320, 486, 3072, 25, 240

Offset: 1

Author

Keith Flower (keith.flower(AT)gmail.com), Nov 28 2009

Keywords

• nonn
• easy

Comments

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.

Examples

			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.
Explanatory table for initial terms:
  n  a(n)   P_{a(n)}
  1    0                (appears here as prescribed)
  2    1                (appears here as prescribed)
                      Calculation of d^2
  3    2 -> [2]       -> 2^2 = 4
  4    4 -> [2,2]     -> 2^2 + 2^2 = 8
  5    3 -> [3]       -> 3^3 = 9
  6    8 -> [2,2,2]   -> 2^2 + 2^2 + 2^2 = 12
  7    6 -> [2,3]     -> 2^2 + 3^2 = 13
  8   16 -> [2,2,2,2] -> 2^2 + 2^2 + 2^2 + 2^2 = 16
  9   12 -> [2,2,3]   -> 2^2 + 2^2 + 3^2 = 17
		

Links

• Keith Flower, Integers in Euclidean space order
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Cf. A001222, A027746, A067666.

Similarly defined sequences: A064364, A178595.

Formula

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

Extensions

Definition edited by N. J. A. Sloane, Nov 29 2009

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

More terms from R. J. Mathar, Jan 25 2010

Edited by Peter Munn, Aug 17 2022