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.

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

0,1

Comments

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?

Links

Table of n, a(n) for n=0..46.

Example

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)).

Prog

(VBA)
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

Crossrefs

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

Sequence in context: A196089 A033359 A326791 * A268854 A317388 A088344

Adjacent sequences: A374867 A374868 A374869 * A374871 A374872 A374873

Keyword

nonn

Author

Lechoslaw Ratajczak, Sep 16 2024

Status

approved