For mathematical definitions see the top-level Addition Chains.
It does seem to be possible to repair the conjecture for some non-Hansen numbers. This page will list the early ones as I try to find a general approach. So far, each chain is hand generated, and the techniques used differ from chain to chain.
See the table in my page on star and \(l^0\) chains for a key to what's been done: Star and L0.
For the first non-Hansen we repair the gap in a way that seems ungeneralizable. We have an addition chain for \(n\) containing a sequence \(a, a+1, a+1+b,2a,2a+c\) and we transform it to a chain for \(2^n-1\) with the sequence:
\(2^a-1,...,2^a(2^a-1),2^{2a}-1,2^{2a}-1+2^a(2^a-1),2^a(2^{a+1}-1),2^{a-b}(2^{a+1+b}-1)\)
The introduction of \(2^a(2^{a+1}-1)\) and \(2^{a-b}(2^{a+1+b}-1)\) into the addition chain will require some latter steps to remove these extra powers of two. Not all chains will be able to do this.
We will use this chain:
1 2 4 8 16 32 64 65 97 128 225 353 706 1412 2824 5648 11296 22592 45184 90368 180736 180801 361537 723074 1446148 2892296 2892393 5784689
| Elements | How First Element Formed | Elements In Row | Chain Length So Far |
| \(1,...,2\) | 2 | 1 | |
| \(2^2-1,...,2^2\cdot (2^2-1)\) | \(2+1\) | 3 | 4 |
| \(2^4-1,...,2^4\cdot (2^4-1)\) | \(2^2\cdot (2^2-1)+2^2-1\) | 5 | 9 |
| \(2^8-1,...,2^8\cdot (2^8-1)\) | \(2^4\cdot (2^4-1)+2^4-1\) | 9 | 18 |
| \(2^{16}-1,...,2^{16}\cdot(2^{16}-1)\) | \(2^8\cdot (2^8-1)+2^8-1\) | 17 | 35 |
| \(2^{32}-1,...,2^1\cdot (2^{32}-1)\) | \(2^{16}\cdot(2^{16}-1)+2^{16}-1\) | 33 | 68 |
| \(2^{64}-1,...,2^{64}\cdot(2^{64}-1))\) | \(2^{32}\cdot(2^{32}-1)+2^{32}-1)\) | 65 | 133 |
| \(2^{128}-1\) | \(2^{64}\cdot(2^{64}-1)+2^{64}-1\) | 1 | 134 |
| \(2^{128}-1+2^{64}\cdot(2^{64}-1)\) | \(2^{128}-1+2^{64}\cdot(2^{64}-1)\) | 1 | 135 |
| \(2^{64}\cdot(2^{65}-1)\) | \(2^{128}-1+2^{64}\cdot(2^{64}-1)+1\) | 1 | 136 |
| \(2^{32}\cdot(2^{97}-1),...,2^{128}\cdot(2^{97}-1)\) | \(2^{64}\cdot(2^{65}-1)+2^{32}\cdot(2^{32}-1)\) | 97 | 233 |
| \(2^{225}-1,...,2^{128}\cdot(2^{225}-1)\) | \(2^{128}\cdot(2^{97}-1)+2^{128}-1\) | 129 | 362 |
| \(2^{353}-1,...,2^{353}\cdot(2^{353}-1)\) | \(2^{128}\cdot(2^{225}-1)+2^{128}-1\) | 354 | 716 |
| \(2^{706}-1,...,2^{706}\cdot2^{706}-1)\) | \(2^{353}\cdot(2^{353}-1)+2^{353}-1\) | 707 | 1423 |
| \(2^{1412}-1,...,2^{1412}\cdot(2^{1412}-1)\) | \(2^{706}\cdot(2^{706}-1)+2^{706}-1\) | 1413 | 2836 |
| \(2^{2824}-1,...,2^{2824}\cdot(2^{2824}-1)\) | \(2^{1412}\cdot(2^{1412}-1)+2^{1412}-1\) | 2825 | 5661 |
| \(2^{5648}-1,...,2^{5648}\cdot(2^{5648}-1)\) | \(2^{2824}\cdot(2^{2824}-1)+2^{2824}-1\) | 5649 | 11310 |
| \(2^{11296}-1,...,2^{11296}\cdot(2^{11296}-1)\) | \(2^{5648}\cdot(2^{5648}-1)+2^{5648}-1\) | 11297 | 22607 |
| \(2^{22592}-1,...,2^{22592}\cdot(2^{22592}-1)\) | \(2^{11296}\cdot(2^{11296}-1)+2^{11296}-1\) | 22593 | 45200 |
| \(2^{45184}-1,...,2^{45184}\cdot(2^{45184}-1)\) | \(2^{22592}\cdot(2^{22592}-1)+2^{22592}-1\) | 45185 | 90385 |
| \(2^{90368}-1,...,2^{90368}\cdot(2^{90368}-1)\) | \(2^{45184}\cdot(2^{45184}-1)+2^{45184}-1\) | 90369 | 180754 |
| \(2^{180736}-1,...,2^{129}\cdot(2^{180736}-1)\) | \(2^{90368}\cdot(2^{90368}-1)+2^{90368}-1\) | 130 | 180884 |
| \(2^{64}\cdot(2^{180801}-1),...,2^{180736}\cdot(2^{180801}-1)\) | \(2^{129}\cdot(2^{180736}-1)+2^{64}\cdot(2^{65}-1)\) | 180673 | 361557 |
| \(2^{361537}-1,...,2^{361537}\cdot(2^{361537}-1)\) | \(2^{180736}\cdot(2^{180801}-1)+2^{180736}-1\) | 361538 | 723095 |
| \(2^{723074}-1,...,2^{723074}\cdot(2^{723074}-1)\) | \(2^{361537}\cdot(2^{361537}-1)+2^{361537}-1\) | 723075 | 1446170 |
| \(2^{1446148}-1,...,2^{1446148}\cdot(2^{1446148}-1)\) | \(2^{723074}\cdot(2^{723074}-1)+2^{723074}-1\) | 1446149 | 2892319 |
| \(2^{2892296}-1,...,2^{129}\cdot(2^{2892296}-1)\) | \(2^{1446148}\cdot(2^{1446148}-1)+2^{1446148}-1\) | 130 | 2892449 |
| \(2^{32}\cdot(2^{2892393}-1),...,2^{2892296}\cdot(2^{2892393}-1)\) | \(2^{129}\cdot(2^{2892296}-1)+2^{32}\cdot(2^{97}-1)\) | 2892265 | 5784714 |
| \(2^{5784689}-1\) | \(2^{2892296}\cdot(2^{2892393}-1)+2^{2892296}-1\) | 1 | 5784715 |
This chain has a length of 5784715. \(l(2^{5784689}-1)\le 5784715=l(5784689)+5784689-1=27+5784689-1=5784715\).
1,
2,
2^2-1,...,2^2*(2^2-1),
2^4-1,...,2^4*(2^4-1),
2^8-1,...,2^8*(2^8-1),
2^16-1,...,2^16*(2^16-1),
2^32-1,...,2^32*(2^32-1),
2^64-1,...,2^64*(2^64-1),
2^128-1,
680564733841876926908302470789826871295, // 2^128-1 + 2^64*(2^64-1)
2^64*(2^65-1),
2^32*(2^97-1),...,2^128*(2^97-1),
2^225-1,...,2^128*(2^225-1),
2^353-1,...,2^353*(2^353-1),
2^706-1,...,2^706*2^706-1),
2^1412-1,...,2^1412*(2^1412-1),
2^2824-1,...,2^2824*(2^2824-1),
2^5648-1,...,2^5648*(2^5648-1),
2^11296-1,...,2^11296*(2^11296-1),
2^22592-1,...,2^22592*(2^22592-1),
2^45184-1,...,2^45184*(2^45184-1),
2^90368-1,...,2^90368*(2^90368-1),
2^180736-1,...,2^129*(2^180736-1),
2^64*(2^180801-1),...,2^180736*(2^180801-1),
2^361537-1,...,2^361537*(2^361537-1),
2^723074-1,...,2^723074*(2^723074-1),
2^1446148-1,...,2^1446148*(2^1446148-1),
2^2892296-1,...,2^129*(2^2892296-1),
2^32*(2^2892393-1),...,2^2892296*(2^2892393-1),
2^5784689-1
For the second non-Hansen we have many chains that don't work but at least one does:
1 2 4 8 16 32 64 65 97 128 225 353 706 1412 2824 5648 11296 22592 45184 90368 180736 361472 361537 723009 1446018 2892036 5784072 5784169 11568241
1,
2,
2^2-1,...,2^2*(2^2-1),
2^4-1,...,2^4*(2^4-1),
2^8-1,...,2^8*(2^8-1),
2^16-1,...,2^16*(2^16-1),
2^32-1,...,2^32*(2^32-1),
2^64-1,...,2^64*(2^64-1),
2^128-1,
680564733841876926908302470789826871295, // 2^128-1 + 2^64*(2^64-1)
2^64*(2^65-1),
2^32*(2^97-1),...,2^128*(2^97-1),
2^225-1,...,2^128*(2^225-1),
2^353-1,...,2^353*(2^353-1),
2^706-1,...,2^706*(2^706-1),
2^1412-1,...,2^1412*(2^1412-1),
2^2824-1,...,2^2824*(2^2824-1),
2^5648-1,...,2^5648*(2^5648-1),
2^11296-1,...,2^11296*(2^11296-1),
2^22592-1,...,2^22592*(2^22592-1),
2^45184-1,...,2^45184*(2^45184-1),
2^90368-1,...,2^90368*(2^90368-1),
2^180736-1,...,2^180736*(2^180736-1),
2^361472-1,...,2^129*(2^361472-1),
2^64*(2^361537-1),...,2^361472*(2^361537-1),
2^723009-1,...,2^723009*(2^723009-1),
2^1446018-1,...,2^1446018*(2^1446018-1),
2^2892036-1,...,2^2892036*(2^2892036-1),
2^5784072-1,...,2^129*(2^5784072-1),
2^32*(2^5784169-1),...,2^5784072*(2^5784169-1),
2^11568241-1
This chain has length 11568268. \(l(2^{11568241}-1)\le 11568268=l(11568241)+11568241-1=28+11568241-1=11568268\).
The 4th non-Hansen:
1,
2,
2^2-1,...,2^2*(2^2-1),
2^4-1,...,2^4*(2^4-1),
2^8-1,...,2^8*(2^8-1),
2^16-1,...,2^16*(2^16-1),
2^32-1,
2^32-1+2^16*(2^16-1),
2^16*(2^17-1),
2^8*(2^25-1),...,2^32*(2^25-1),
2^57-1,...,2^32*(2^57-1),
2^89-1,...,2^89*(2^89-1),
2^178-1,...,2^178*(2^178-1),
2^356-1,...,2^356*(2^356-1),
2^712-1,...,2^712*(2^712-1),
2^1424-1,...,2^1424*(2^1424-1),
2^2848-1,...,2^2848*(2^2848-1),
2^5696-1,...,2^5696*(2^5696-1),
2^11392-1,...,2^11392*(2^11392-1),
2^22784-1,...,2^33*(2^22784-1),
2^16*(2^22801-1),...,2^22784*(2^22801-1),
2^45585-1,...,2^45585*(2^45585-1),
2^91170-1,...,2^91170*(2^91170-1),
2^182340-1,...,2^182340*(2^182340-1),
2^364680-1,...,2^364680*(2^364680-1),
2^729360-1,...,2^729360*(2^729360-1),
2^1458720-1,...,2^1458720*(2^1458720-1),
2^2917440-1,...,2^2917440*(2^2917440-1),
2^5834880-1,...,2^33*(2^5834880-1),
2^8*(2^5834905-1),...,2^5834880*(2^5834905-1),
2^11669785-1
This chain has length 11669812. \(l(2^{11669785}-1)\le 11669812=l(11669785)+11669785-1=28+11669785-1=11669812\).
For the 5th non-Hansen:
1,
2,
2^2-1,...,2^2*(2^2-1),
2^4-1,...,2^4*(2^4-1),
2^8-1,...,2^8*(2^8-1),
2^16-1,...,2^16*(2^16-1),
2^32-1,
2^32-1+2^16*(2^16-1),
2^16*(2^17-1),
2^8*(2^25-1),...,2^32*(2^25-1),
2^57-1,...,2^32*(2^57-1),
2^89-1,...,2^89*(2^89-1),
2^178-1,...,2^178*(2^178-1),
2^356-1,...,2^356*(2^356-1),
2^712-1,...,2^712*(2^712-1),
2^1424-1,...,2^1424*(2^1424-1),
2^2848-1,...,2^2848*(2^2848-1),
2^5696-1,...,2^5696*(2^5696-1),
2^11392-1,...,2^11392*(2^11392-1),
2^22784-1,...,2^33*(2^22784-1),
2^8*(2^22809-1),...,2^22784*(2^22809-1),
2^45593-1,...,2^45593*(2^45593-1),
2^91186-1,...,2^91186*(2^91186-1),
2^182372-1,...,2^182372*(2^182372-1),
2^364744-1,...,2^364744*(2^364744-1),
2^729488-1,...,2^729488*(2^729488-1),
2^1458976-1,...,2^1458976*(2^1458976-1),
2^2917952-1,...,2^2917952*(2^2917952-1),
2^5835904-1,...,2^33*(2^5835904-1),
2^16*(2^5835921-1),...,2^5835904*(2^5835921-1),
2^11671825-1
This chain has length 11671852. \(l(2^{11671825}-1)\le 11671852=l(11671825)+11671825-1=28+11671825-1=11671852\).
For the 6th non-Hansen:
1,
2,
2^2-1,...,2^2*(2^2-1),
2^4-1,...,2^4*(2^4-1),
2^8-1,...,2^8*(2^8-1),
2^16-1,...,2^16*(2^16-1),
2^32-1,
2^32-1+2^16*(2^16-1),
2^16*(2^17-1),
2^8*(2^25-1),...,2^32*(2^25-1),
2^57-1,...,2^32*(2^57-1),
2^89-1,...,2^89*(2^89-1),
2^178-1,...,2^178*(2^178-1),
2^356-1,...,2^356*(2^356-1),
2^712-1,...,2^712*(2^712-1),
2^1424-1,...,2^1424*(2^1424-1),
2^2848-1,...,2^2848*(2^2848-1),
2^5696-1,...,2^33*(2^5696-1),
2^16*(2^5713-1),...,2^5696*(2^5713-1),
2^11409-1,...,2^11409*(2^11409-1),
2^22818-1,...,2^22818*(2^22818-1),
2^45636-1,...,2^45636*(2^45636-1),
2^91272-1,...,2^91272*(2^91272-1),
2^182544-1,...,2^182544*(2^182544-1),
2^365088-1,...,2^365088*(2^365088-1),
2^730176-1,...,2^730176*(2^730176-1),
2^1460352-1,...,2^1460352*(2^1460352-1),
2^2920704-1,...,2^2920704*(2^2920704-1),
2^5841408-1,...,2^33*(2^5841408-1),
2^8*(2^5841433-1),...,2^5841408*(2^5841433-1),
2^11682841-1,
This chain has length 11682868. \(l(2^{11682841}-1)\le 11682868=l(11682841)+11682841-1=28+11682841-1=11682868\).
For the 7th non-Hansen (23097633) with \(l(23097633)=29\) we use a different technique. We have an addition chain for \(n\) containing a sequence \(a, a+b, a+b+c,2a\) and we transform it to a chain for \(2^n-1\) with the sequence:
\(2^a-1,...,2^a(2^a-1),2^{a-b}(2^{a+b}-1),2^{a-b-c}(2^{a+b+c}-1),2^{2a}-1\). For this to work we must have \(a\ge b+c,2b\ge a, b+2c\ge a\).
We show below a chain of length 23097661.
| Elements | How First Element Formed | Elements In Row | Chain Length So Far |
| \(1,...,2\) | 2 | 1 | |
| \(2^2-1,...,2^2\cdot (2^2-1)\) | \(2+1\) | 3 | 4 |
| \(2^4-1,...,2^4\cdot (2^4-1)\) | \(2^2\cdot (2^2-1)+2^2-1\) | 5 | 9 |
| \(2^8-1,...,2^8\cdot (2^8-1)\) | \(2^4\cdot (2^4-1)+2^4-1\) | 9 | 18 |
| \(2^{16}-1,...,2^{16}\cdot(2^{16}-1)\) | \(2^8\cdot (2^8-1)+2^8-1\) | 17 | 35 |
| \(2^{32}-1,...,2^1\cdot (2^{32}-1)\) | \(2^{16}\cdot(2^{16}-1)+2^{16}-1\) | 2 | 37 |
| \(2^2\cdot (2^{32}-1),...,2^{32}\cdot (2^{32}-1)\) | \(2^1\cdot (2^{32}-1)+2^1\cdot (2^{32}-1)\) | 31 | 68 |
| \(2^{64}-1,...,2^{64}\cdot (2^{64}-1)\) | \(2^{32}\cdot (2^{32}-1)+2^{32}-1\) | 65 | 133 |
| \(2^{128}-1,...,2^{128}\cdot (2^{128}-1)\) | \(2^{64}\cdot (2^{64}-1)+2^{64}-1\) | 129 | 262 |
| \(2^{256}-1,...,2^{256}\cdot(2^{256}-1)\) | \(2^{128}\cdot (2^{128}-1)+2^{128}-1\) | 257 | 519 |
| \(2^{512}-1,...,2^{512}\cdot(2^{512}-1)\) | \(2^{256}\cdot(2^{256}-1)+2^{256}-1\) | 513 | 1032 |
| \(2^{1024}-1,...,2^{1024}\cdot(2^{1024}-1)\) | \(2^{512}\cdot(2^{512}-1)+2^{512}-1\) | 1025 | 2057 |
| \(2^{2048}-1,...,2^{2048}\cdot(2^{2048}-1)\) | \(2^{1024}\cdot(2^{1024}-1)+2^{1024}-1\) | 2049 | 4106 |
| \(2^{4096}-1,...,2^{4096}\cdot(2^{4096}-1)\) | \(2^{2048}\cdot(2^{2048}-1)+2^{2048}-1\) | 4097 | 8203 |
| \(2^{8192}-1,...,2^{8192}\cdot(2^{8192}-1)\) | \(2^{4096}\cdot(2^{4096}-1)+2^{4096}-1\) | 8193 | 16396 |
| \(2^{16384}-1,...,2^{33}\cdot(2^{16384}-1)\) | \(2^{8192}\cdot(2^{8192}-1)+2^{8192}-1\) | 34 | 16430 |
| \(2^1\cdot(2^{16416}-1)\) | \(2^{33}\cdot(2^{16384}-1)+2^1\cdot (2^{32}-1)\) | 1 | 16431 |
| \(2^{16417}-1\) | \(2^1\cdot(2^{16416}-1)+1\) | 1 | 16432 |
| \(2^{34}\cdot(2^{16384}-1),...,2^{16384}\cdot(2^{16384}-1)\) | \(2^{33}\cdot(2^{16384}-1)+2^{33}\cdot(2^{16384}-1)\) | 16351 | 32783 |
| \(2^{32768}-1,...,2^{32768}\cdot(2^{32768}-1),\) | \(2^{16384}\cdot(2^{16384}-1)+2^{16384}-1\) | 32769 | 65552 |
| \(2^{65536}-1,...,2^{16417}\cdot(2^{65536}-1)\) | \(2^{32768}\cdot(2^{32768}-1)+2^{32768}-1\) | 16418 | 81970 |
| \(2^{81953}-1,...,2^{81953}\cdot(2^{81953}-1)\) | \(2^{81953}\cdot(2^{81953}-1)+2^{81953}-1\) | 81954 | 163924 |
| \(2^{163906}-1,...,2^{16417}\cdot(2^{163906}-1)\) | \(2^{81953}\cdot(2^{81953}-1)+2^{81953}-1\) | 16418 | 180342 |
| \(2^1\cdot(2^{180322}-1),...,2^{180323}\cdot(2^{180322}-1)\) | \(2^{16417}\cdot(2^{163906}-1)+2^1\cdot(2^{16416}-1)\) | 180323 | 360665 |
| \(2^1\cdot(2^{360644}-1),...,2^{360645}\cdot(2^{360644}-1)\) | \(2^{180323}\cdot(2^{180322}-1)+2^1\cdot(2^{180322}-1)\) | 360645 | 721310 |
| \(2^1\cdot(2^{721288}-1),...,2^{721289}\cdot(2^{721288}-1)\) | \(2^{360645}\cdot(2^{360644}-1)+2^1\cdot(2^{360644}-1)\) | 721289 | 1442599 |
| \(2^1\cdot(2^{1442576}-1),...,2^{1442577}\cdot(2^{1442576}-1)\) | \(2^{721289}\cdot(2^{721288}-1)+2^1\cdot(2^{721288}-1)\) | 1442577 | 2885176 |
| \(2^1\cdot(2^{2885152}-1),...,2^{2885153}\cdot(2^{2885152}-1)\) | \(2^{1442577}\cdot(2^{1442576}-1)+2^1\cdot(2^{1442576}-1)\) | 2885153 | 5770329 |
| \(2^1\cdot(2^{5770304}-1),...,2^{5770305}\cdot(2^{5770304}-1)\) | \(2^{2885153}\cdot(2^{2885152}-1)+2^1\cdot(2^{2885152}-1)\) | 5770305 | 11540634 |
| \(2^1\cdot(2^{11540608}-1),...,2^{11540609}\cdot(2^{11540608}-1)\) | \(2^{5770305}\cdot(2^{5770304}-1)+2^1\cdot(2^{5770304}-1)\) | 11540609 | 23081243 |
| \(2^1\cdot(2^{23081216}-1),...,2^{16417}\cdot(2^{23081216}-1)\) | \(2^{11540609}\cdot(2^{11540608}-1)+2^1\cdot(2^{11540608}-1)\) | 16417 | 23097660 |
| \(2^{23097633}-1\) | \(2^{16417}\cdot(2^{23081216}-1)+2^{16417}-1\) | 1 | 23097661 |
The conjecture says that \(l(2^{23097633}-1)\le l(23097633)+23097633-1=23097661\).
In the machine-readable format:
1,
2,
2^2-1,...,2^2*(2^2-1),
2^4-1,...,2^4*(2^4-1),
2^8-1,...,2^8*(2^8-1),
2^16-1,...,2^16*(2^16-1),
2^32-1,...,2^1*(2^32-1),
2^2*(2^32-1),...,2^32*(2^32-1),
2^64-1,...,2^64*(2^64-1),
2^128-1,...,2^128*(2^128-1),
2^256-1,...,2^256*(2^256-1),
2^512-1,...,2^512*(2^512-1),
2^1024-1,...,2^1024*(2^1024-1),
2^2048-1,...,2^2048*(2^2048-1),
2^4096-1,...,2^4096*(2^4096-1),
2^8192-1,...,2^8192*(2^8192-1),
2^16384-1,...,2^33*(2^16384-1),
2^1*(2^16416-1),
2^16417-1,
2^34*(2^16384-1),...,2^16384*(2^16384-1),
2^32768-1,...,2^32768*(2^32768-1),
2^65536-1,...,2^16417*(2^65536-1),
2^81953-1,...,2^81953*(2^81953-1),
2^163906-1,...,2^16417*(2^163906-1),
2^1*(2^180322-1),...,2^180323*(2^180322-1),
2^1*(2^360644-1),...,2^360645*(2^360644-1),
2^1*(2^721288-1),...,2^721289*(2^721288-1),
2^1*(2^1442576-1),...,2^1442577*(2^1442576-1),
2^1*(2^2885152-1),...,2^2885153*(2^2885152-1),
2^1*(2^5770304-1),...,2^5770305*(2^5770304-1),
2^1*(2^11540608-1),...,2^11540609*(2^11540608-1),
2^1*(2^23081216-1),...,2^16417*(2^23081216-1),
2^23097633-1
We can also repair the chain for \(2^{31942247}-1\):
1,
2,
2^2-1,...,2^2*(2^2-1),
2^4-1,...,2^2*(2^4-1),
2^6-1,...,2^1*(2^6-1)
2^2*(2^6-1),...,2^6*(2^6-1)
2^12-1,...,2^12*(2^12-1),
2^24-1,...,2^24*(2^24-1),
2^48-1,...,2^48*(2^48-1),
2^96-1,...,2^7*(2^96-1),
2^1*(2^102-1),
2^103-1,
2^8*(2^96-1),...,2^96*(2^96-1),
2^192-1,...,2^192*(2^192-1),
2^384-1,...,2^103*(2^384-1),
2^487-1,...,2^487*(2^487-1),
2^974-1,...,2^974*(2^974-1)
2^1948-1,...,2^1948*(2^1948-1),
2^3896-1,...,2^3896*(2^3896-1),
2^7792-1,...,2^7792*(2^7792-1),
2^15584-1,...,2^15584*(2^15584-1),
2^31168-1,...,2^31168*(2^31168-1),
2^62336-1,...,2^62336*(2^62336-1),
2^124672-1,...,2^103*(2^124672-1),
2^1*(2^124774-1),...,2^124775*(2^124774-1),
2^1*(2^249548-1),...,2^249549*(2^249548-1)
2^1*(2^499096-1),...,2^499097*(2^499096-1),
2^1*(2^998192-1),...,2^998193*(2^998192-1),
2^1*(2^1996384-1),...,2^1996385*(2^1996384-1),
2^1*(2^3992768-1),...,2^3992769*(2^3992768-1),
2^1*(2^7985536-1),...,2^7985537*(2^7985536-1),
2^1*(2^15971072-1),...,2^15971073*(2^15971072-1),
2^1*(2^31942144-1),...,2^103*(2^31942144-1),
2^31942247-1
This chain has length 31942276. \(l(2^{31942247}-1)\le 31942276\le l(31942247)+31942247-1=30+31942247-1=31942276\).
We can also repair the chain for \(2^{32364653}-1\):
1,
2,
2^2-1,...,2^2*(2^2-1),
2^4-1,...,2^2*(2^4-1),
2^6-1,...,2^6*(2^6-1),
2^12-1,...,2^1*(2^12-1),
2^2*(2^12-1),...,2^12*(2^12-1),
2^24-1,...,2^24*(2^24-1),
2^48-1,...,2^48*(2^48-1),
2^96-1,...,2^13*(2^96-1),
2^1*(2^108-1),
2^109-1,
2^14*(2^96-1),...,2^96*(2^96-1),
2^192-1,...,2^192*(2^192-1),
2^384-1,...,2^109*(2^384-1),
2^493-1,...,2^493*(2^493-1),
2^986-1,...,2^986*(2^986-1),
2^1972-1,...,2^1972*(2^1972-1),
2^3944-1,...,2^3944*(2^3944-1),
2^7888-1,...,2^7888*(2^7888-1),
2^15776-1,...,2^15776*(2^15776-1),
2^31552-1,...,2^31552*(2^31552-1),
2^63104-1,...,2^109*(2^63104-1),
2^1*(2^63212-1),...,2^63213*(2^63212-1),
2^1*(2^126424-1),...,2^126425*(2^126424-1),
2^1*(2^252848-1),...,2^252849*(2^252848-1),
2^1*(2^505696-1),...,2^505697*(2^505696-1),
2^1*(2^1011392-1),...,2^1011393*(2^1011392-1),
2^1*(2^2022784-1),...,2^2022785*(2^2022784-1),
2^1*(2^4045568-1),...,2^4045569*(2^4045568-1),
2^1*(2^8091136-1),...,2^8091137*(2^8091136-1),
2^1*(2^16182272-1),...,2^16182273*(2^16182272-1),
2^1*(2^32364544-1),...,2^109*(2^32364544-1),
2^32364653-1
This chain has length 32364682. \(l(2^{32364653}-1)\le 32364682\le l(32364653)+32364653-1=30+32364653-1=32364682\).
The 9th non-Hansen:
1,
2,
2^2-1,...,2^2*(2^2-1),
2^4-1,...,2^4*(2^4-1),
2^8-1,...,2^8*(2^8-1),
2^16-1,...,2^16*(2^16-1),
2^32-1,...,2^32*(2^32-1),
2^64-1,...,2^64*(2^64-1),
2^128-1,
680564733841876926908302470789826871295, // 2^128 - 1 + 2^64 * (2^64 - 1)
2^64*(2^65-1),
2^32*(2^97-1),...,2^128*(2^97-1),
2^225-1,...,2^128*(2^225-1),
2^353-1,...,2^353*(2^353-1),
2^706-1,...,2^706*(2^706-1),
2^1412-1,...,2^1412*2^1412-1),
2^2824-1,...,2^2824*(2^2824-1),
2^5648-1,...,2^5648*(2^5648-1),
2^11296-1,...,2^11296*(2^11296-1),
2^22592-1,...,2^22592*(2^22592-1),
2^45184-1,...,2^45184*(2^45184-1),
2^90368-1,...,2^90368*(2^90368-1),
2^180736-1,...,2^180736*(2^180736-1),
2^361472-1,...,2^361472*(2^361472-1),
2^722944-1,...,2^129*(2^722944-1),
2^64*(2^723009-1),...,2^722944*(2^723009-1),
2^1445953-1,...,2^1445953*(2^1445953-1),
2^2891906-1,...,2^2891906*(2^2891906-1),
2^5783812-1,...,2^5783812*(2^5783812-1),
2^11567624-1,...,2^129*(2^11567624-1),
2^32*(2^11567721-1),...,2^11567624*(2^11567721-1),
2^23135345-1
This chain has length 23135373. \(l(2^{23135345}-1)\le 23135373=l(23135345)+23135345-1=29+23135345-1=23135373\).
The 12th non-Hansen:
1,
2,
2^2-1,...,2^2*(2^2-1),
2^4-1,...,2^4*(2^4-1),
2^8-1,...,2^8*(2^8-1),
2^16-1,...,2^16*(2^16-1),
2^32-1,...,2^32*(2^32-1),
2^64-1,...,2^64*(2^64-1),
2^128-1,...,2^128*(2^128-1),
2^256-1,...,2^256*(2^256-1),
2^512-1,...,2^512*(2^512-1),
2^1024-1,...,2^1024*(2^1024-1),
2^2048-1,
2^2048-1+2^1024 (2^1024-1),
2^1024*(2^1025-1),
2^512*(2^1537-1),...,2^2048*(2^1537-1),
2^3585-1,...,2^2048*(2^3585-1),
2^5633-1,...,2^5633*(2^5633-1),
2^11266-1,...,2^11266*(2^11266-1),
2^22532-1,...,2^22532*(2^22532-1),
2^45064-1,...,2^45064*(2^45064-1),
2^90128-1,...,2^90128*(2^90128-1),
2^180256-1,...,2^2049*(2^180256-1),
2^1024*(2^181281-1),...,2^180256*(2^181281-1),
2^361537-1,...,2^361537*(2^361537-1),
2^723074-1,...,2^723074*(2^723074-1),
2^1446148-1,...,2^1446148*(2^1446148-1),
2^2892296-1,...,2^2892296*(2^2892296-1),
2^5784592-1,...,2^5784592*(2^5784592-1),
2^11569184-1,...,2^2049*(2^11569184-1),
2^512*(2^11570721-1),...,2^11569184*(2^11570721-1),
2^23139905-1
This chain has length 23139933. \(l(2^{23139905}-1)\le 23139933=l(23139905)+23139905-1=29+23139905-1=23139933\).
The 13th non-Hansen:
1,
2,
2^2-1,...,2^2*(2^2-1),
2^4-1,...,2^4*(2^4-1),
2^8-1,...,2^8*(2^8-1),
2^16-1,...,2^16*(2^16-1),
2^32-1,...,2^32*(2^32-1),
2^64-1,
36893488143124135935, // 2^64-1 + 2^32*(2^32-1)
2^32*(2^33-1),
2^16*(2^49-1),...,2^64*(2^49-1)
2^113-1,...,2^64*(2^113-1),
2^177-1,...,2^177*(2^177-1),
2^354-1,...,2^354*(2^354-1),
2^708-1,...,2^708*(2^708-1),
2^1416-1,...,2^1416*(2^1416-1),
2^2832-1,...,2^2832*(2^2832-1),
2^5664-1,...,2^5664*(2^5664-1),
2^11328-1,...,2^65*(2^11328-1),
2^32*(2^11361-1),...,2^11328*(2^11361-1),
2^22689-1,...,2^22689*(2^22689-1),
2^45378-1,...,2^45378*(2^45378-1),
2^90756-1,...,2^90756**(2^90756-1),
2^181512-1,...,2^181512*(2^181512-1),
2^363024-1,...,2^363024*(2^363024-1),
2^726048-1,...,2^726048*(2^726048-1),
2^1452096-1,...,2^1452096*(2^1452096-1),
2^2904192-1,...,2^2904192*(2^2904192-1),
2^5808384-1,...,2^5808384*(2^5808384-1),
2^11616768-1,...,2^65*(2^11616768-1),
2^16*(2^11616817-1),...,2^11616768*(2^11616817-1),
2^23233585-1,
This chain has length 23233613. \(l(2^{23233585}-1)\le 23233613=l(23233585)+23233585-1=29+23233585-1=23233613\).
The 14th non-Hansen:
1,
2,
2^2-1,...,2^2*(2^2-1),
2^4-1,...,2^4*(2^4-1),
2^8-1,...,2^8*(2^8-1),
2^16-1,...,2^16*(2^16-1),
2^32-1,
2^32-1+2^16*(2^16-1),
2^16*(2^17-1),
2^8*(2^25-1),...,2^32*(2^25-1),
2^57-1,...,2^32*(2^57-1),
2^89-1,...,2^89*(2^89-1),
2^178-1,...,2^178*(2^178-1),
2^356-1,...,2^356*(2^356-1),
2^712-1,...,2^712*(2^712-1),
2^1424-1,...,2^1424*(2^1424-1),
2^2848-1,...,2^2848*(2^2848-1),
2^5696-1,...,2^5696*(2^5696-1),
2^11392-1,...,2^11392*(2^11392-1),
2^22784-1,...,2^33*(2^22784-1),
2^16*(2^22801-1),...,2^22784*(2^22801-1),
2^45585-1,...,2^45585*(2^45585-1),
2^91170-1,...,2^91170*(2^91170-1),
2^182340-1,...,2^182340*(2^182340-1),
2^364680-1,...,2^364680*(2^364680-1),
2^729360-1,...,2^729360*(2^729360-1),
2^1458720-1,...,2^1458720*(2^1458720-1),
2^2917440-1,...,2^2917440*(2^2917440-1),
2^5834880-1,...,2^5834880*(2^5834880-1),
2^11669760-1,...,2^33*(2^11669760-1),
2^8*(2^11669785-1),...,2^11669760*(2^11669785-1),
2^23339545-1
This chain has length 23339573. \(l(2^{23339545}-1)\le 23339573=l(23339545)+23339545-1=29+23339545-1=23339573\).
The 8th non-Hansen:
1,
2,
2^2-1,...,2^2*(2^2-1),
2^4-1,...,2^4*(2^4-1),
2^8-1,...,2^8*(2^8-1),
2^16-1,...,2^16*(2^16-1),
2^32-1,...,2^32*(2^32-1),
2^64-1,...,2^64*(2^64-1),
2^128-1,...,2^128*(2^128-1),
2^256-1,
2^256-1+2^128*(2^128-1),
2^128*(2^129-1),
2^64*(2^193-1),...,2^256*(2^193-1),
2^449-1,...,2^256*(2^449-1),
2^705-1,...,2^705*2^705-1),
2^1410-1,...,2^1410*(2^1410-1),
2^2820-1,...,2^2820*(2^2820-1),
2^5640-1,...,2^5640*(2^5640-1),
2^11280-1,...,2^11280*(2^11280-1),
2^22560-1,...,2^22560*(2^22560-1),
2^45120-1,...,2^45120*(2^45120-1),
2^90240-1,...,2^90240*(2^90240-1),
2^180480-1,...,2^180480*(2^180480-1),
2^360960-1,...,2^257*(2^360960-1),
2^128*(2^361089-1),...,2^360960*(2^361089-1),
2^722049-1,...,2^722049*(2^722049-1),
2^1444098-1,...,2^1444098*(2^1444098-1),
2^2888196-1,...,2^2888196*(2^2888196-1),
2^5776392-1,...,2^5776392*(2^5776392-1),
2^11552784-1,...,2^257*(2^11552784-1),
2^64*(2^11552977-1),...,2^11552784*(2^11552977-1),
2^23105761-1,
This chain has length 23105789. \(l(2^{23105761}-1)\le 23105789=l(23105761)+23105761-1=29+23105761-1=23105789\).
The 16th non-Hansen:
1,
2,
2^2-1,...,2^2*(2^2-1),
2^4-1,...,2^4*(2^4-1),
2^8-1,...,2^8*(2^8-1),
2^16-1,...,2^16*(2^16-1),
2^32-1,
8589869055, // 2^32-1+2^16*(2^16-1),
2^16*(2^17-1),
2^8*(2^25-1),...,2^32*(2^25-1),
2^57-1,...,2^32*(2^57-1),
2^89-1,...,2^89*(2^89-1),
2^178-1,...,2^178*(2^178-1),
2^356-1,...,2^356*(2^356-1),
2^712-1,...,2^712*(2^712-1),
2^1424-1,...,2^1424*(2^1424-1),
2^2848-1,...,2^2848*(2^2848-1),
2^5696-1,...,2^5696*(2^5696-1),
2^11392-1,...,2^11392*(2^11392-1),
2^22784-1,...,2^33*(2^22784-1),
2^8*(2^22809-1),...,2^22784*(2^22809-1),
2^45593-1,...,2^45593*(2^45593-1),
2^91186-1,...,2^91186*(2^91186-1),
2^182372-1,...,2^182372*(2^182372-1),
2^364744-1,...,2^364744*(2^364744-1),
2^729488-1,...,2^729488*(2^729488-1),
2^1458976-1,...,2^1458976*(2^1458976-1),
2^2917952-1,...,2^2917952*(2^2917952-1),
2^5835904-1,...,2^5835904*(2^5835904-1),
2^11671808-1,...,2^33*(2^11671808-1),
2^16*(2^11671825-1),...,2^11671808*(2^11671825-1),
2^23343633-1,
This chain has length 23343661. \(l(2^{23343633}-1)\le 23343661=l(23343633)+23343633-1=29+23343633-1=23343661\).
The 27th non-Hansen:
1,
2,
2^2-1,...,2^2(2^2-1),
2^4-1,...,2^4(2^4-1),
2^8-1,...,2^8(2^8-1),
2^16-1,...,2^1(2^16-1),
2^2(2^16-1),...,2^16(2^16-1),
2^32-1,...,2^32(2^32-1),
2^64-1,...,2^64(2^64-1),
2^128-1,...,2^128(2^128-1),
2^256-1,...,2^256(2^256-1),
2^512-1,...,2^512(2^512-1),
2^1024-1,...,2^1024(2^1024-1),
2^2048-1,...,2^2048(2^2048-1),
2^4096-1,...,2^4096(2^4096-1),
2^8192-1,...,2^8192(2^8192-1),
2^16384-1,...,2^17(2^16384-1),
2^18(2^16384-1),...,2^16384(2^16384-1),
2^1(2^16400-1),
2^16401-1,
2^32768-1,...,2^16401(2^32768-1),
2^16402(2^32768-1),...,2^49169(2^32768-1),
2^49169-1,
2^81937-1,...,2^16401(2^81937-1),
2^1(2^98337-1),...,2^81937(2^98337-1),
2^180274-1,...,2^180274(2^180274-1),
2^360548-1,...,2^360548(2^360548-1),
2^721096-1,...,2^721096(2^721096-1),
2^1442192-1,...,2^1442192(2^1442192-1),
2^2884384-1,...,2^2884384(2^2884384-1),
2^5768768-1,...,2^5768768(2^5768768-1),
2^11537536-1,...,2^11537536(2^11537536-1),
2^23075072-1,...,2^16401(2^23075072-1),
2^16402(2^23075072-1),...,2^23091473(2^23075072-1),
2^23091473-1,
2^46166545-1,
This chain has length 46166574. \(l(2^{46166545}-1)\le 46166574=l(46166545)+46166545-1=30+46166545-1=46166574\).