For mathematical definitions see the top-level Addition Chains.
This is the smallest example I have so far.
| Elements | How First Element Formed | Elements In Row | Chain Length So Far |
| \(1\) | 1 | 0 | |
| \(2\) | \(1+1\) | 1 | 1 |
| \(3\) | \(2+1\) | 1 | 2 |
| \(6,...,24\) | \(3+3\) | 3 | 5 |
| \(2^1\cdot(2^4-1),...,2^5\cdot(2^4-1)\) | \(24+6\) | 5 | 10 |
| \(483\) | \(2^5\cdot (2^4 - 1)+3\) | 1 | 11 |
| \(963,...,2^3\cdot 963\) | \(483 + 2^5\cdot (2^4-1)\) | 4 | 15 |
| \(8187\) | \(2^3\cdot 963 + 483\) | 1 | 16 |
| \(8189\) | \(8187 + 2\) | 1 | 17 |
| \(2^3\cdot(2^{11}-1),...,2^{13}\cdot(2^{11}-1)\) | \(8189 + 8187\) | 11 | 28 |
| \(16777213\) | \(2^{13}\cdot (2^{11}-1) + 8189\) | 1 | 29 |
| \(2^{24}-1\) | \(16777213 + 2\) | 1 | 30 |
| \(2^2\cdot(2^{23}-1),...,2^{24}\cdot(2^{23}-1)\) | \(2^{24}-1 + 16777213\) | 23 | 53 |
| \(2^{47}-1,...,2^{47}\cdot(2^{47}-1)\) | \(2^{24} \cdot (2^{23}-1)+2^{24}-1\) | 48 | 101 |
| \(2^{94}-1,...,2^{94}\cdot(2^{94}-1)\) | \(2^{47}\cdot(2^{47}-1)+2^{47}-1\) | 95 | 196 |
| \(2^{188}-1,...,2^{188}\cdot(2^{188}-1)\) | \(2^{94}\cdot(2^{94}-1)+2^{94}-1\) | 189 | 385 |
| \(2^{376}-1,...,2^{376}\cdot(2^{376}-1)\) | \(2^{188}\cdot(2^{188}-1)+2^{188}-1\) | 377 | 762 |
| \(2^{752}-1,...,2^{752}\cdot(2^{752}-1)\) | \(2^{376}\cdot(2^{376}-1)+2^{376}-1\) | 753 | 1515 |
| \(2^{1504}-1,...,2^{1504}\cdot(2^{1504}-1)\) | \(2^{752}\cdot(2^{752}-1)+2^{752}-1\) | 1505 | 3020 |
| \(2^{3008}-1,...,2^{3008}\cdot(2^{3008}-1)\) | \(2^{1504}\cdot(2^{1504}-1)+2^{1504}-1\) | 3009 | 6029 |
| \(2^{6016}-1,...,2^{14}\cdot(2^{6016}-1)\) | \(2^{3008}\cdot(2^{3008}-1)+2^{3008}-1\) | 15 | 6044 |
| \(2^3\cdot(2^{6027}-1),...,2^{6016}\cdot(2^{6027}-1)\) | \(2^{14}\cdot(2^{6016}-1)+2^3\cdot(2^{11}-1)\) | 6014 | 12058 |
| \(2^{12043}-1,...,2^{12043}\cdot(2^{12043}-1)\) | \(2^{6016}\cdot(2^{6027}-1)+2^{6016}-1\) | 12044 | 24102 |
| \(2^{24086}-1,...,2^{24086}\cdot(2^{24086}-1)\) | \(2^{12043}\cdot(2^{12043}-1)+2^{12043}-1\) | 24087 | 48189 |
| \(2^{48172}-1,...,2^{48172}\cdot(2^{48172}-1)\) | \(2^{24086}\cdot(2^{24086}-1)+2^{24086}-1\) | 48173 | 96362 |
| \(2^{96344}-1,...,2^{96344}\cdot(2^{96344}-1)\) | \(2^{48172}\cdot(2^{48172}-1)+2^{48172}-1\) | 96345 | 192707 |
| \(2^{192688}-1,...,2^{192688}\cdot(2^{192688}-1)\) | \(2^{96344}\cdot(2^{96344}-1)+2^{96344}-1\) | 192689 | 385396 |
| \(2^{385376}-1,...,2^{385376}\cdot(2^{385376}-1)\) | \(2^{192688}\cdot(2^{192688}-1)+2^{192688}-1\) | 385377 | 770773 |
| \(2^{770752}-1,...,2^{25}\cdot(2^{770752}-1)\) | \(2^{385376}\cdot(2^{385376}-1)+2^{385376}-1\) | 26 | 770799 |
| \(2^2\cdot(2^{770775}-1),...,2^{770752}\cdot(2^{770775}-1)\) | \(2^{25}\cdot(2^{770752}-1)+2^2\cdot(2^{23}-1)\) | 770751 | 1541550 |
| \(2^{1541527}-1\) | \(2^{770752}\cdot(2^{770775}-1)+2^{770752}-1\) | 1 | 1541551 |
So we have \(l(2^{1541527}-1)\le 1541551 < l(1541527)+1541527-1=26+1541527-1=1541552\).
This chain in a machine-readable format:
1,
2,
3,
6,...,24,
2^1*(2^4-1),...,2^5*(2^4-1),
483,
963,...,7704,
8187,
8189,
2^3*(2^11-1),...,2^13*(2^11-1),
16777213,
2^24-1,
2^2*(2^23-1),...,2^24*(2^23-1),
2^47-1,...,2^47*(2^47-1),
2^94-1,...,2^94*(2^94-1),
2^188-1,...,2^188*(2^188-1),
2^376-1,...,2^376*(2^376-1),
2^752-1,...,2^752*(2^752-1),
2^1504-1,...,2^1504*(2^1504-1),
2^3008-1,...,2^3008*(2^3008-1),
2^6016-1,...,2^14*(2^6016-1),
2^3*(2^6027-1),...,2^6016*(2^6027-1),
2^12043-1,...,2^12043*(2^12043-1),
2^24086-1,...,2^24086*(2^24086-1),
2^48172-1,...,2^48172*(2^48172-1),
2^96344-1,...,2^96344*(2^96344-1),
2^192688-1,...,2^192688*(2^192688-1),
2^385376-1,...,2^385376*(2^385376-1),
2^770752-1,...,2^25*(2^770752-1),
2^2*(2^770775-1),...,2^770752*(2^770775-1),
2^1541527-1,