A381497 a(n) = sum of numbers k < n such that 1 < gcd(k,n) and rad(k) != rad(n), where rad = A007947.
0, 0, 0, 0, 0, 9, 0, 6, 6, 25, 0, 36, 0, 49, 45, 42, 0, 81, 0, 100, 84, 121, 0, 144, 45, 169, 96, 196, 0, 315, 0, 210, 198, 289, 175, 354, 0, 361, 273, 430, 0, 609, 0, 484, 435, 529, 0, 648, 140, 655, 459, 676, 0, 801, 385, 826, 570, 841, 0, 1260, 0, 961, 798
Offset: 1
Keywords
Examples
Table of n and a(n) for select n, showing prime power decomposition of both and row n of A381094:
n Factor(n) a(n) Factor(a(n)) Row n of A381094
-------------------------------------------------------------------
6 2 * 3 9 3^2 {2,3,4}
8 2^3 6 2 * 3 {6}
9 3^2 6 2 * 3 {6}
10 2 * 5 25 5^2 {2,4,5,6,8}
12 2^2 * 3 36 2^2 * 3^2 {2,3,4,8,9,10}
14 2 * 7 49 7^2 {2,4,6,7,8,10,12}
15 3 * 5 45 3^2 * 5 {3,5,6,9,10,12}
16 2^4 42 2 * 3 * 7 {6,10,12,14}
18 2 * 3^2 81 3^4 {2,3,4,8,9,10,14,15,16}
20 2^2 * 5 100 2^2 * 5^2 {2,4,5,6,8,12,14,15,16,18}
21 3 * 7 84 2^2 * 3 * 7 {3,6,7,9,12,14,15,18}
22 2 * 11 121 11^2 {2,4,6,8,10,11,12,14,16,18,20}
24 2^3 * 3 144 2^4 * 3^2 {2,3,4,8,9,10,14,15,16,20,21,22}
a(6) = (2+4) + (3) = 9,
a(n) = 6 for n in {8, 9} since 6 is the only number less than n that shares a factor with n but does not have the same squarefree kernel as n.
a(10) = (2+4+6+8) + (5) = 25.
a(12) = (2+4+8+10) + (3+9) = 36.
a(14) = (2+4+6+8+10+12) + (7) = 49.
a(15) = (3+6+9+12) + (5+10) = 45.
a(16) = (6+10+12+14) = 42, etc.
Links
- Michael De Vlieger, Table of n, a(n) for n = 1..16384
- Michael De Vlieger, Log log scatterplot of a(n), n = 6..2^14, ignoring a(n) = 0, showing a(n) for prime power n in gold, a(n) for squarefree n in green, otherwise blue.
Crossrefs
Programs
-
Mathematica
rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]]; Table[r = rad[n]; Total@ Select[Range[n], Nor[CoprimeQ[#, n], rad[#] == r] &], {n, 120}]
Comments