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

Author

Lechoslaw Ratajczak, Sep 16 2024

Keywords

• nonn

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?

Examples

			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)).
		

Crossrefs

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

Programs

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