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

1,1

Comments

a(14) > 3*10^6 (if it exists). Is there any even term?

Links

Table of n, a(n) for n=1..13.

Example

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.

Prog

(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)))));

Crossrefs

Cf. A048158, A375595, A378275.

Sequence in context: A145619 A027490 A393548 * A323688 A266453 A279116

Adjacent sequences: A380150 A380151 A380152 * A380154 A380155 A380156

Keyword

nonn,more

Author

Lechoslaw Ratajczak, Jan 13 2025

Extensions

a(9)-a(13) from Jinyuan Wang, Jan 14 2025

Status

approved