login
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.
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.
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
Similarly defined sequences: A064364, A178595.
Sequence in context: A124833 A181815 A324931 * A244982 A101468 A188866
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