A381499 a(n) = sum of numbers k < n such that 1 < gcd(k,n) < k and rad(k) does not divide n, where rad = A007947.
0, 0, 0, 0, 0, 0, 0, 6, 6, 6, 0, 10, 0, 28, 28, 42, 0, 39, 0, 65, 65, 80, 0, 102, 45, 126, 96, 159, 0, 111, 0, 210, 148, 210, 138, 253, 0, 280, 221, 338, 0, 342, 0, 411, 366, 444, 0, 547, 140, 563, 403, 601, 0, 700, 344, 708, 512, 750, 0, 751, 0, 868, 703, 930
Offset: 1
Keywords
Examples
Table of n and a(n) for select n, showing prime power decomposition of the latter and row n of A272619:
n a(n) Factor(a(n)) Row n of A272619
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8 6 2 * 3 {6}
9 6 2 * 3 {6}
10 6 2 * 3 {6}
12 10 2 * 5 {10}
14 28 2^2 * 7 {6,10,12}
15 28 2^2 * 7 {6,10,12}
16 42 2 * 3 * 7 {6,10,12,14}
18 39 3 * 13 {10,14,15}
20 65 5 * 13 {6,12,14,15,18}
21 65 5 * 13 {6,12,14,15,18}
22 80 2^4 * 5 {6,10,12,14,18,20}
24 102 2 * 3 * 17 {10,14,15,20,21,22}
25 45 3^2 * 5 {10,15,20}
26 126 2 * 3^2 * 7 {6,10,12,14,18,20,22,24}
27 96 2^5 * 3 {6,12,15,18,21,24}
28 159 3 * 53 {6,10,12,18,20,21,22,24,26}
Links
- Michael De Vlieger, Table of n, a(n) for n = 1..16384
- Michael De Vlieger, Log log scatterplot of a(n), n = 8..2^14, ignoring a(n) = 0, showing a(n) for prime power n in gold, a(n) for squarefree n in green, otherwise blue.
Programs
-
Mathematica
rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]]; Table[r = rad[n]; If[PrimeQ[n], 0, Total@ Select[Range[n], And[1 < GCD[#, n] < #, ! Divisible[n, rad[#]]] &]], {n, 120}]
Formula
a(n) is the sum of row n of A272619.
a(n) = 0 for prime n, n = 4, and n = 6.
Comments