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

1,3

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.

Links

Table of n, a(n) for n=1..61.

Keith Flower, Integers in Euclidean space order

Index entries for sequences that are permutations of the natural numbers

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

Example

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

Crossrefs

Cf. A001222, A027746, A067666.

Similarly defined sequences: A064364, A178595.

Sequence in context: A124833 A181815 A324931 * A244982 A101468 A188866

Adjacent sequences: A168518 A168519 A168520 * A168522 A168523 A168524

Keyword

nonn,easy

Author

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

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

Status

approved