A375595
Numbers m for which the sum of all values of k satisfying the equation: m mod k = floor((m - k)/k) mod k (1 <= k <= m) exceeds 2*m.
Original entry on oeis.org
23, 29, 35, 41, 47, 53, 59, 65, 71, 77, 79, 83, 89, 95, 99, 101, 107, 111, 113, 119, 125, 131, 137, 139, 143, 149, 155, 159, 161, 167, 173, 179, 185, 191, 197, 199, 203, 209, 215, 219, 221, 223, 227, 233, 239, 245, 251, 257, 259, 263, 269
Offset: 1
Let T(i,j) be the triangle read by rows: T(i,j) = 1 if i mod j = floor((i - j)/j) mod j, T(i,j) = 0 otherwise, for 1 <= j <= i. The triangle begins:
i\j| 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ...
-----------------------------------------
1| 1
2| 1 1
3| 1 0 1
4| 1 0 0 1
5| 1 1 0 0 1
6| 1 1 0 0 0 1
7| 1 0 1 0 0 0 1
8| 1 0 0 0 0 0 0 1
9| 1 1 0 1 0 0 0 0 1
10| 1 1 0 0 0 0 0 0 0 1
11| 1 0 1 0 1 0 0 0 0 0 1
12| 1 0 1 0 0 0 0 0 0 0 0 1
13| 1 1 0 0 0 1 0 0 0 0 0 0 1
14| 1 1 0 1 0 0 0 0 0 0 0 0 0 1
15| 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1
...
The j-th column has period j^2. Consecutive elements of this period are j X j identity matrix entries, read by rows.
11 is not in this sequence because only k's <= 11 satisfying the equation 11 mod k = floor((11 - k)/k) mod k are: 1, 3, 5, 11, hence 1+3+5+11 = 20 and 20 < 2*11.
23 is in this sequence because only k's <= 23 satisfying the equation 23 mod k = floor((23 - k)/k) mod k are: 1, 5, 7, 11, 23, hence 1+5+7+11+23 = 47 and 47 > 2*23.
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(f(i,j):=mod(i-floor((i-j)/j),j),
(n:0, for m:2 thru 500 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)))));
A378275
Numbers m which satisfy the equation: (m - floor((m - k)/k)) mod k = 1 (1 <= k <= m) only for k = 2 and m - 1.
Original entry on oeis.org
3, 4, 7, 11, 19, 23, 59, 83, 167, 227, 491, 659, 839, 983, 1019, 1091, 1319, 1459, 1523, 1847, 2179, 2503, 2963, 3719, 3767, 4519, 4871, 4919, 5059, 6563, 9239, 9419, 10883, 12107, 12539, 14891, 15383, 20071, 20747, 23819, 25219, 26759, 33851, 35591, 37379, 45191
Offset: 1
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 ...
-----+------------------
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
...
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).
7 is in this sequence because the only k's satisfying the equation (7 - floor((7 - k)/k)) mod k = 1 are 2 and (7-1).
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(f(i, j):=mod((i-floor((i-j)/j)), j),
(n:3, for t:7 thru 100000 step 4 do
(for k:3 while f(t, k)#1 and k
-
is(m) = if(m%4==3, for(k=3, m\2, if((m-m\k)%k==0, return(0))); 1, m==4); \\ Jinyuan Wang, Jan 14 2025
A374870
Let e(m) be the sum of all values of k satisfying the equation: (m mod k = floor((m - k)/k) mod k), minus 2*m (1 <= k <= m); then a(n) is the smallest m for which e(m) = n, or 0 if no e(m) has value n.
Original entry on oeis.org
39, 23, 5847, 735, 65, 29, 35, 77, 111, 173, 415, 185, 79, 47, 113, 137, 317, 867, 307, 543, 4843, 2153, 1203, 161, 59, 159, 351, 531, 1577, 475, 617, 89, 5321, 95, 11405, 1371, 107, 83, 219, 197, 199, 1855, 365, 6521, 3667, 8597, 131
Offset: 0
Let T(i,j) be the triangle read by rows: T(i,j) = 1 if i mod j = floor((i - j)/j) mod j, T(i,j) = 0 otherwise, for 1 <= j <= i.
The triangle begins:
i\j | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ...
-----------------------------------------
1 | 1
2 | 1 1
3 | 1 0 1
4 | 1 0 0 1
5 | 1 1 0 0 1
6 | 1 1 0 0 0 1
7 | 1 0 1 0 0 0 1
8 | 1 0 0 0 0 0 0 1
9 | 1 1 0 1 0 0 0 0 1
10 | 1 1 0 0 0 0 0 0 0 1
11 | 1 0 1 0 1 0 0 0 0 0 1
12 | 1 0 1 0 0 0 0 0 0 0 0 1
13 | 1 1 0 0 0 1 0 0 0 0 0 0 1
14 | 1 1 0 1 0 0 0 0 0 0 0 0 0 1
15 | 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1
...
The j-th column has period j^2. Consecutive elements of this period are j X j identity matrix entries, read by rows.
a(0) = 39 because 39 is the smallest m for which e(m) = 0 (only k's satisfying the equation: 39 mod k = floor((39 - k)/k) mod k are: 1, 3, 7, 9, 19, 39, hence: 1+3+7+9+19+39-2*39 = 0 = e(39)).
a(2) = 5847 because 5847 is the smallest m for which e(m) = 2 (only k's satisfying the equation: 5847 mod k = floor((5847 - k)/k) mod k are: 1, 85, 135, 171, 343, 730, 1461, 2923, 5847, hence: 1+85+135+171+343+730+1461+2923+5847-2*5847 = 2 = e(5847)).
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Sub calcul()
For m = 1 To 500000
s = 0
For k = 1 To WorksheetFunction.Floor(m / 2, 1)
If (m - WorksheetFunction.Floor((m - k) / k, 1)) Mod k = 0 Then
s = s + k
End If
Next k
If s > m Then
e = s - m
v = WorksheetFunction.Ceiling(e / 1000000, 1)
If IsEmpty(Cells(e - (v - 1) * 1000000, v)) = False Then
Else
Cells(e - (v - 1) * 1000000, v).Value = m
End If
End If
Next m
End Sub
Showing 1-3 of 3 results.
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