Repairing the Scholz-Brauer Conjecture

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.
 

Update (September 2026): Claude Fable 5.1 generalized one of the techniques I used and gave it the "spawn" name. It wrote code to fix 6 of the problematic entries. The spawn technique used for the first non-Hansen number does generalize, and it can be searched for mechanically. Take a chain for \(n\) containing \(a,a+1,c=a+1+b,2a\) and continuing \(2a+c,(2a+c)+2a\). Form \(2^{2a}=(2^{2a}-1)+1\) (one extra addition, but it raises the top bit like a doubling), then \(2^a(2^{a+1}-1)=2^a(2^a-1)+2^{2a}\) and \(2^{a-b}(2^c-1)=2^a(2^{a+1}-1)+2^{a-b}(2^b-1)\). The elements \(a+1\) and \(c\) are born pre-shifted and may be reused later at any shift not below their birth shift. The construction reaches \(l(n)+n-1\) exactly when \(a-b\) does not exceed the number of doublings applied to \(b\) (so \(b\ge a/2\) when \(b\) lies on the doubling run) and \(2a\) is used only unshifted, i.e. the chain does not double \(2a\) again. Modelling every element as \(2^s(2^v-1)\) with a free offset \(s\) plus at most two such spawned powers of two as a constraint problem (OR-Tools CP-SAT, all 272 optimal chains of the twelve unrepaired numbers below 46467170) finds the 29th, 30th, 32nd, 37th, 38th and 39th non-Hansen numbers, listed at the end of this page, and proves that the other six (33167549, 33247629, 33573479, 33580601, 33626279, 44177985) cannot be repaired this way: each has \(a-b\) too large for \(b\), or doubles \(2a\) onward. The six constructions were verified by computing every element of the chain for \(2^n-1\) exactly.

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

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.

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.  \(l(2^{46166545}-1)\le 46166574=l(46166545)+46166545-1=30+46166545-1=46166574\).

The 29th non-Hansen (46205137) uses the spawn template with \(a=128\):

1 2 4 8 16 32 64 128 129 193 256 449 705 1410 2820 5640 11280 22560 45120 90240 180480 360960 721920 1443840 1443969 2887809 5775618 11551236 23102472 23102665 46205137

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)+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,...,2^360960*(2^360960-1),
2^721920-1,...,2^721920*(2^721920-1),
2^1443840-1,...,2^257*(2^1443840-1),
2^128*(2^1443969-1),...,2^1443840*(2^1443969-1),
2^2887809-1,...,2^2887809*(2^2887809-1),
2^5775618-1,...,2^5775618*(2^5775618-1),
2^11551236-1,...,2^11551236*(2^11551236-1),
2^23102472-1,...,2^257*(2^23102472-1),
2^64*(2^23102665-1),...,2^23102472*(2^23102665-1),
2^46205137-1

This chain has length 46205166. \(l(2^{46205137}-1)\le 46205166=l(46205137)+46205137-1=30+46205137-1=46205166\).

The 30th non-Hansen (46207201) uses the spawn template with \(a=128\):

1 2 4 8 16 32 64 128 129 193 256 449 705 1410 2820 5640 11280 22560 45120 90240 180480 360960 721920 722049 1443969 2887938 5775876 11551752 23103504 23103697 46207201

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)+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,...,2^360960*(2^360960-1),
2^721920-1,...,2^257*(2^721920-1),
2^128*(2^722049-1),...,2^721920*(2^722049-1),
2^1443969-1,...,2^1443969*(2^1443969-1),
2^2887938-1,...,2^2887938*(2^2887938-1),
2^5775876-1,...,2^5775876*(2^5775876-1),
2^11551752-1,...,2^11551752*(2^11551752-1),
2^23103504-1,...,2^257*(2^23103504-1),
2^64*(2^23103697-1),...,2^23103504*(2^23103697-1),
2^46207201-1

This chain has length 46207230. \(l(2^{46207201}-1)\le 46207230=l(46207201)+46207201-1=30+46207201-1=46207230\).

The 32nd non-Hansen (46269553) uses the spawn template with \(a=64\):

1 2 4 8 16 32 64 65 97 128 225 353 706 1412 2824 5648 11296 22592 45184 90368 180736 361472 722944 1445888 1445953 2891841 5783682 11567364 23134728 23134825 46269553

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)+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,...,2^722944*(2^722944-1),
2^1445888-1,...,2^129*(2^1445888-1),
2^64*(2^1445953-1),...,2^1445888*(2^1445953-1),
2^2891841-1,...,2^2891841*(2^2891841-1),
2^5783682-1,...,2^5783682*(2^5783682-1),
2^11567364-1,...,2^11567364*(2^11567364-1),
2^23134728-1,...,2^129*(2^23134728-1),
2^32*(2^23134825-1),...,2^23134728*(2^23134825-1),
2^46269553-1

This chain has length 46269582. \(l(2^{46269553}-1)\le 46269582=l(46269553)+46269553-1=30+46269553-1=46269582\).

The 37th non-Hansen (46416433) uses the spawn template with \(a=32\):

1 2 4 8 16 32 33 49 64 113 177 354 708 1416 2832 5664 11328 22656 45312 45345 90657 181314 362628 725256 1450512 2901024 5802048 11604096 23208192 23208241 46416433

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)+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,...,2^11328*(2^11328-1),
2^22656-1,...,2^22656*(2^22656-1),
2^45312-1,...,2^65*(2^45312-1),
2^32*(2^45345-1),...,2^45312*(2^45345-1),
2^90657-1,...,2^90657*(2^90657-1),
2^181314-1,...,2^181314*(2^181314-1),
2^362628-1,...,2^362628*(2^362628-1),
2^725256-1,...,2^725256*(2^725256-1),
2^1450512-1,...,2^1450512*(2^1450512-1),
2^2901024-1,...,2^2901024*(2^2901024-1),
2^5802048-1,...,2^5802048*(2^5802048-1),
2^11604096-1,...,2^11604096*(2^11604096-1),
2^23208192-1,...,2^65*(2^23208192-1),
2^16*(2^23208241-1),...,2^23208192*(2^23208241-1),
2^46416433-1

This chain has length 46416462. \(l(2^{46416433}-1)\le 46416462=l(46416433)+46416433-1=30+46416433-1=46416462\).

The 38th non-Hansen (46433329) uses the spawn template with \(a=32\):

1 2 4 8 16 32 33 49 64 113 177 354 708 1416 2832 5664 11328 22656 22689 45345 90690 181380 362760 725520 1451040 2902080 5804160 11608320 23216640 23216689 46433329

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)+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,...,2^11328*(2^11328-1),
2^22656-1,...,2^65*(2^22656-1),
2^32*(2^22689-1),...,2^22656*(2^22689-1),
2^45345-1,...,2^45345*(2^45345-1),
2^90690-1,...,2^90690*(2^90690-1),
2^181380-1,...,2^181380*(2^181380-1),
2^362760-1,...,2^362760*(2^362760-1),
2^725520-1,...,2^725520*(2^725520-1),
2^1451040-1,...,2^1451040*(2^1451040-1),
2^2902080-1,...,2^2902080*(2^2902080-1),
2^5804160-1,...,2^5804160*(2^5804160-1),
2^11608320-1,...,2^11608320*(2^11608320-1),
2^23216640-1,...,2^113*(2^23216640-1),
2^64*(2^23216689-1),...,2^23216640*(2^23216689-1),
2^46433329-1

This chain has length 46433358. \(l(2^{46433329}-1)\le 46433358=l(46433329)+46433329-1=30+46433329-1=46433358\).

The 39th non-Hansen (46467121) uses the spawn template with \(a=32\):

1 2 4 8 16 32 33 49 64 113 177 354 708 1416 2832 5664 11328 11361 22689 45378 90756 181512 363024 726048 1452096 2904192 5808384 11616768 23233536 23233585 46467121

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)+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,...,2^11616768*(2^11616768-1),
2^23233536-1,...,2^113*(2^23233536-1),
2^64*(2^23233585-1),...,2^23233536*(2^23233585-1),
2^46467121-1

This chain has length 46467150. \(l(2^{46467121}-1)\le 46467150=l(46467121)+46467121-1=30+46467121-1=46467150\).