Euler's numeri idonei or idoneal numbers (suitable or convenient numbers) were included in four papers that Euler presented to the Petersberg Academy in 1778. They are:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 15, 16, 18, 21, 22, 24, 25, 28, 30, 33, 37, 40, 42, 45, 48, 57, 58, 60, 70, 72, 78, 85, 88, 93, 102, 105, 112, 120, 130, 133, 165, 168, 177, 190, 210, 232, 240, 253, 273, 280, 312, 330, 345, 357, 385, 408, 462, 520, 760, 840, 1320, 1365, 1848
The property that these numbers share is that they cannot be represented in the form ab+ac+bc with 0<a<b<c. While 1848 is the largest known number, this source states that:
S. Chowla proved (in 1934) that the number of numeri idonei is finite, and it is known that there can be at most ONE more square-free numerus idoneus beyond those found by Euler. Whether such another number exists is still an open question.
However, another source states that 1848 is the largest number if the Riemann Hypothesis hold true otherwise what's said above will hold.
I came across the reference to these numbers when investigating the number associated with my diurnal age today, namely 27358. It is a member of OEIS
A094377 | Greatest number having exactly n representations as ab+ac+bc with 0<a<b<c. |
- 5 - 134 - 192
- 14 - 32 - 585
- 19 - 34 - 504
- 23 - 56 - 330
- 26 - 81 - 236
- 26 - 105 - 188
- 29 - 134 - 144
- 30 - 41 - 368
- 30 - 112 - 169
- 34 - 72 - 235
- 35 - 66 - 248
- 44 - 53 - 258
- 44 - 107 - 150
- 56 - 102 - 137
- 57 - 70 - 184
- 66 - 91 - 136
- 71 - 108 - 110
- 9 - 14 - 1184
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