A380153 Numbers m for which the sum of all values of k satisfying the equation: (m - floor((m - k)/k)) mod k = 0 (1 <= k <= m) equals 2*m.
39, 4395, 29055, 57931, 81115, 152571, 164955, 410731, 664747, 877435, 2080875, 2521087, 2539515
Offset: 1
Examples
Let T(i,j) be the triangle read by rows: T(i,j) = (i - floor((i - j)/j)) mod j for 1 <= j <= i. The triangle begins: i\j | 1 2 3 4 5 6 7 8 9 10 11 ... -----+------------------------ 1 | 0 2 | 0 0 3 | 0 1 0 4 | 0 1 1 0 5 | 0 0 2 1 0 6 | 0 0 2 2 1 0 7 | 0 1 0 3 2 1 0 8 | 0 1 1 3 3 2 1 0 9 | 0 0 1 0 4 3 2 1 0 10 | 0 0 2 1 4 4 3 2 1 0 11 | 0 1 0 2 0 5 4 3 2 1 0 ... The j-th column has period j^2, r-th element of this period has the form (r - 1 - floor((r - 1)/j)) mod j (1 <= r <= j^2). The period of j-th column consists of the sequence (0,1,2,...,j-1) and its consecutive j-1 right rotations (moving rightmost element to the left end). 39 is in this sequence because the only k's <= 39 satisfying the equation (39 - floor((39 - k)/k)) mod k = 0 are: 1, 3, 7, 9, 19, 39, hence: 1+3+7+9+19+39 = 78 = 2*39.
Programs
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Maxima
(f(i, j):=mod(i-floor((i-j)/j), j), (n:0, for m:2 thru 5000 do (s:0, for k:1 thru floor(m/2) do (if f(m, k)=0 then (s:s+k)), if s=m then (n:n+1, print(n , "" , m)))));
Extensions
a(9)-a(13) from Jinyuan Wang, Jan 14 2025
Comments