A375595 -id:A375595 - cp's OEIS frontend

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.

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

Lechoslaw Ratajczak, Nov 21 2024

nonn

Every term greater than 4 has the form 4*t + 3.
Let b(z) be the number of elements of this sequence <= z:
-------------
   z  | b(z)
-------------
 10^2 |   8
 10^3 |  14
 10^4 |  32
 10^5 |  55
 10^6 | 125
 10^7 | 347
 10^8 | 950
-------------
Every term greater than 4 is prime.

			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).

Jinyuan Wang, Table of n, a(n) for n = 1..3000

Cf. A000040, A048158, A051731, A375007, A375595.

(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

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.

Original entry on oeis.org

39, 4395, 29055, 57931, 81115, 152571, 164955, 410731, 664747, 877435, 2080875, 2521087, 2539515

Offset: 1

Lechoslaw Ratajczak, Jan 13 2025

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

			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.

Cf. A048158, A375595, A378275.

(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)))));

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

A380305 Triangle read by rows: T(n,k) = (n - floor((n - k)/k)) mod k, for 0 < k <= n.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 2, 1, 0, 0, 0, 2, 2, 1, 0, 0, 1, 0, 3, 2, 1, 0, 0, 1, 1, 3, 3, 2, 1, 0, 0, 0, 1, 0, 4, 3, 2, 1, 0, 0, 0, 2, 1, 4, 4, 3, 2, 1, 0, 0, 1, 0, 2, 0, 5, 4, 3, 2, 1, 0, 0, 1, 0, 2, 1, 5, 5, 4, 3, 2, 1, 0, 0, 0, 1, 3, 2, 0, 6, 5, 4, 3, 2, 1, 0

Offset: 1

Lechoslaw Ratajczak, Jan 19 2025

The triangle is a variant of the triangle in A048158. The period of the k-th column consists of the period of the k-th column in the triangle in A048158 (0,1,2,...,k-1) and its consecutive k-1 right rotations (moving the rightmost element to the left end). Thus the k-th column has period k^2 and the r-th element of this period has the form (r - 1 - floor((r - 1)/k)) mod k (1 <= r <= k^2).

Such as the triangle in A048158 may be the basis for definitions of different kinds of numbers (abundant numbers, perfect numbers, etc.), this triangle may be the basis for definitions of counterparts of these numbers (elements of A375595 as counterparts of abundant numbers, elements of A380153 as counterparts of perfect numbers).

			Triangle begins:
n\k| 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
 ...

Cf. A048158, A375595, A378275, A380153.

T[n_,k_]:=Mod[n - Floor[(n - k)/k], k]; Table[T[n,k], {n,13},{k,n}]//Flatten (* Stefano Spezia, Jan 20 2025 *)

(for n:1 thru 25 do
(for k:1 thru n do
(T[n,k]:mod(n-floor((n-k)/k),k)),
print(makelist(T[n,i], i, 1, n))));

row(n) = vector(n, k, (n - floor((n - k)/k)) % k); \\ Michel Marcus, Jan 20 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

Lechoslaw Ratajczak, Sep 16 2024

nonn

The three smallest values of n (n_1, n_2, n_3) for which a(n) is unknown after computing consecutive e(t) for 1 <= t <= z:
    z    |   n_1   |   n_2   |   n_3   |
----------------------------------------
10^5     |   309   |   343   |   352   |
2*10^5   |   394   |   556   |   558   |
3*10^5   |   647   |   706   |   755   |
4*10^5   |   941   |   951   |   962   |
5*10^5   |   951   |   964   |  1069   |
Are there any values of n for which a(n) = 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)).

Cf. A005101, A033880, A051731, A294347, A375007, A375595.

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

A381060 Numbers t which are the sum of some subset of the values of k satisfying the equation (t - floor((t - k)/k)) mod k = 0 (t > 1, 1 <= k < t).

Original entry on oeis.org

23, 29, 39, 41, 53, 59, 65, 71, 77, 79, 83, 89, 99, 101, 107, 111, 113, 119, 125, 137, 143, 149, 155, 161, 167, 173, 179, 185, 191, 197, 199, 209, 221, 227, 233, 239, 245, 251, 257, 263, 269, 279, 281, 287, 293, 299, 305, 311, 317, 323, 329, 335, 339, 341, 349, 353, 359, 365, 371

Offset: 1

Lechoslaw Ratajczak, Feb 12 2025

nonn

The sequence is based on the triangle in A380305, which is a variant of the triangle in A048158. Thus, the elements of this sequence are counterparts of pseudoperfect numbers (A005835), such as the elements of A375595 are counterparts of abundant numbers (A005101).
The sequence includes all elements of A380153.
The first even element of this sequence is a(768) = 4094.

			23 is in this sequence because the only k's < 23 satisfying the equation (23 - floor((23 - k)/k)) mod k = 0 are: 1, 5, 7, 11, hence: 5+7+11 = 23.
29 is in this sequence because the only k's < 29 satisfying the equation (29 - floor((29 - k)/k)) mod k = 0 are: 1, 2, 3, 5, 9, 14, hence: 1+2+3+9+14 = 29 and 1+5+9+14 = 29.
47 is not in this sequence because the only k's < 47 satisfying the equation (47 - floor((47 - k)/k)) mod k = 0  are: 1, 3, 7, 11, 15, 23 and no subset of these numbers adds to 47.

Cf. A005835, A048158, A375595, A380153, A380305.

(kill(all), s(y):=(f(i,j):=mod(i-floor((i-j)/j),j),s:0,x:1,
      for k:1 thru floor(y/2) do
              (if f(y,k)=0 then
              (s:s+k, B[x]:k, x:x+1)),
B:setify(makelist(B[r],r,1,x-1)), s),
n:1, for t:2 thru 1000 do
              (if s(t)>=t  then
                     (for b:2 while b<=x-1 and e#t do
                     (C:args(powerset(B,b)),
                            for h:1 while h<=length(C) and e#t do
                            (e:apply("+" , args(C[h])),
                                    if e=t then
                                    (print(n , " " , t), n:n+1))))));

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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.

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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.

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A380305 Triangle read by rows: T(n,k) = (n - floor((n - k)/k)) mod k, for 0 < k <= n.

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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.

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A381060 Numbers t which are the sum of some subset of the values of k satisfying the equation (t - floor((t - k)/k)) mod k = 0 (t > 1, 1 <= k < t).

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