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
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
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
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