Ideas on generating odd k such that σ(k)/k > 4?

[Jianing Song]

Hello everyone. I was working on A100796, the number of all extensions over Q_2 with degree n in the algebraic closure of Q_2. There is a formula which is rather complicated, but the only part needed is:
σ(k) * 2^(2*m) * (2^((2^m-1)*k) - 2^((2^m-2)*k)) < A100976(2^m * k) ≤ σ(k) * 2^(2*m) * 2^((2^m-1)*k), k odd.

The interesting part is, since σ(k)/k is not bounded, we have
Claim. for every m, there exists an odd k such that A100976(2^m * k) > A100976(2^m * (k+2)).
Proof. There exists k' such that σ(k')/k' >= (k'+3)/(k'*(1-2^(-k'))) * 2^(2*(2^m-1)). Then every multiple of k also satisfies this inequality. Take k to be an odd multiple of k' such that k+2 is prime (which exists by Dirichlet's theorem on arithmetic progressions), then k answers the question.

On the other hand, note that the RHS of the inequality involving k' has doubly exponential growth. A heuristic argument shows that min{k : σ(k)/k ≥ m} also has doubly exponential growth, so min{k odd : A100976(2^m * k) > A100976(2^m * (k+2))} is likely to have quadruply(!) exponential growth.

So only the m=1 case merit being added to the OEIS, where A100976(2*k) = σ(k)*(2^(k+2) - 1) for odd k. I'd like to propose
Odd numbers k such that A100976(2*k) > A100976(2*(k+2)).
In other words, these are odd numbers k such that σ(k) > σ(k+2)*(2^(k+4) - 1)/(2^(k+2) - 1). Of course, we need to search among odd k such that σ(k)/k > 4. Note that the smallest such k0 is k0 = 1853070540093840001956842537745897243375. Indeed, we have A100976(2*k) > A100976(2*(k+2)) for k = 5*k0, 7*k0, 9*k0, 11*k0, 17*k0, 19*k0, 23*k0, 27*k0, 29*k0, 31*k0, 33*k0, 39*k0, 41*k0, ...

I was wondering if anybody could calculate some first odd numbers k such that σ(k)/k > 4. Thank you very much in advance!

[Paul Hanna]

Jianing, 
      Just an observation that may or may not be helpful. 
It appears that the generating functions  
F(x) for A100976 - OEIS (offset 1) and
G(x) for A131139 - OEIS ("2-wild partitions" - offset 0)
are related by the logarithmic derivative 
   F(x) = x*G'(x)/G(x). 
Equivalently, 
   G(x) = exp( Integral F(x)/x dx ). 
 
If the above holds true, then the product formula given in the Roberts link in A131139 may shed light on A100976 and consequently on your question. 
But then again, the above observation may only be a coincidence of initial terms. 
 
- Paul 

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[Allan Wechsler]

I have in my head a fairly straightforward depth-first search algorithm that will yield all the odd numbers k with abundancy greater than 4 less than some limit N. It won't yield them in ascending order, and you have to specify N in advance, but certainly it will find all the numbers k that satisfy the constraint, and with nowhere near the expense of simply checking each number.

Is this algorithm obvious, that is, is it well-known to everybody who travels these waters, or is it worth my sketching it out in greater detail?

-- Allan

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[Alex Violette]

Hi Jianing, 

I just so happen to have some odd k that satisfy that constraint lying around in one of my spreadsheets I obtained while trying to hunt for numbers where σ(k)-nk is positive and within a certain bound that I have converted into two lists for you:

σ(k)/k > 4

28653392588170406505341237671920748619590125,1661623794535952315007686817731787979978125,242071729759998279081413044462473388430625,1853070540093840001956842537745897243375,10860627271287767743971868567756435705125,5320030348672254562476052286444417821521411375,1650453007278489284930103705576879531243395094375,30859951052155454708779761513788784173833560844997551871325,5320076120839568708787520437993189804450052125,6408291014834700478176804841836253757623761161004819225,2513075465915553072587475009581219002651875,6436588468472692183875689272766617167637125,27748198008684686051223192149055692101816058120255556375,263817009286091472160531845335754431875125,33323704367819273021779070662889159701708759725,19402161896044932103065918262812133480623749909636581373005588673758888352625,2239081748599958718866433291614639924255625,18288323406152096615209584797401193913331386947840787370266393675,63131165020330765552667449874112622773967108425846100597425,621594413440761589187607182043176599657125,1488295781019085787475624039623629023500625,1320409242484345866151941581259121406847706420678843246125,8636068527390334352381003657808651899252009658009979423922025,8032852581133752156797299272753436921126875,896540044254322112060865708612693239274375,10744248390079249298381878668971867688658057371681902114440970325,157367094625482157955621387833269408070941054897547972979175,1152217793609159443742128616241970014061291997568400008107475,28975190090679719722651120489137630890390478541591486160025,373303170294217561317758534468406864126185299013826166953383775,10876569458254428833303839444352002265485875,5611295823189757671496131041876310590473023939415025262197482321919133261868875,11495362521262925051975671042993205271834375,11139244257867465180302368063978146350625,189749434042796828595032649006478207600146246777733632701575575,669930202672491067748345187565360351715066625,7911204658682312327390376900272836752990375,92916591382574180034274121038457549995283133375,1366714920639445753189931277165828918970875,1094894298033961386327150736714270762673343926689632589910068793364375,25946723360890053696367815832222624980078393129489166188439575,735794933943601726977080692839655988402625,31430338501584003950911503412497931697773871034525,33630396726685436460498140217313418912923437529838593781025,31430338846848789354697303105060729981466405934675,36369299365660403010990250723488603628544625,106856756351262073257629188671224275198254121083017581192224675,31430340918437501777412101260437519683621615335575,410353468529873655398985083371243081882421154398646031914170925,342193504047841011233602301134891437077812045296751639011825,31430342644761428796341099723251511102084289836325,5894346083873629203119220218020420705634625,31430344025820570411484298493502704236854429436925,6433694347067377085445666788353733125875,3184526014338119206059488766093410969445375,8078346207890837581406005050855250345461066039478388289305133975,174492253569132699362445642414829251980126625,31430348168997995256913894804256283641164848238725,31430348859527566064485494189381880208549918039025,11458499856535317764909164905687177336375,31430349895321922275842893267070275059627522739475,1026417475695757430052231916344157254614625,47619673420261750877415116027855141168625,194072702294546996159996958363252901185375,228714557567249440845158942278546368755470875,229544949858095359355842489851995066822383780444406110200537675
σ(k)/k > 5
541241312669435437347268804215330697893940761326502763438750289807079921020765071225422988448494447313597439975005843129310598125,1083030436836249659198303729298459668423341409835989522023996455688334438903683344017407994367257799070910414450632302334345974375,326097996034073476728267918483900546722405265059526060519412603711870320121814791884433669617612838573704825029723887018125,1125297149265505185059229248942411826274372251653250601991918083285561123149757978103975627746847998315000379709049654216875,158686889546222270551603372445601690257490850132071247982385346411853736982025729007259010878869856627483940208518569351875,45454370665658723709685990669481699355919392750285081321023831159666370321532963243384035629263565709607752902046185132060401784202710934090916685625,2159790145701431758590834870146925233736076893472311474048133893200384408473801392025197038430247947667959214509458825311875,1634417674640332627739077334250186362110546553887361987342041131093930366822091714287761581362511555805818344458135300941875,32765231030368130505321066844937242590760815444336460510630176237887770948319561245495290354102667740411619243107358060501875,1089790142278439034319911491486927400603643332613175715693216533550714977724428713919473650871239286586606735468933642760748125,6600773824300260813196120602548504864109208467235174817679435588568246090660160721786581097503613624214215111206253857979375,527619585819749859848471862896747858220754776070893932822299292582372894442042225327384653499780281422067541270922752249375,25901997161768905853888764965493050286667335706371836401840811952895576592476598913995329782080452802846683794542890340039375,58348068789702541542540337556989734284405072814056808320424990206864405474468685568873748608608466827071760350234947394107813125,525856562575427462874568054130983516903335039448807697581550236686873060738244158826897701642466448973357434218104543358125,344895121559870543744997735684352013142664615231697288050276638511372144215546695148618293017174706839808268571282056635625
As for how I found them, for each k I started with a seed where  σ(k)/k is "close" to some integer n but does not exceed it and then I try to multiply primes onto it to take it past n. Hope this helps!

Best,
Alex Violette

To view this discussion visit https://groups.google.com/d/msgid/seqfan/CADy-sGGYrfGoD1_Mw5vSLVxarDFKkjEmJh3208N_RMwJfzFUjw%40mail.gmail.com.

[David desJardins]

On Tue, Jan 6, 2026 at 8:31 PM Allan Wechsler <acw...@gmail.com> wrote:

I have in my head a fairly straightforward depth-first search algorithm that will yield all the odd numbers k with abundancy greater than 4 less than some limit N. It won't yield them in ascending order, and you have to specify N in advance, but certainly it will find all the numbers k that satisfy the constraint, and with nowhere near the expense of simply checking each number.

I think that's largely right. I think it should be possible to generate all of the 4-abundant odd numbers less than N in time roughly proportional to the length of the list of results (times some log factors). It's kind of messy to set the thresholds properly, though. The prime powers are annoying to deal with. It's more work than I want to do right now. 

  -- David desJardins

[Jianing Song]

Thank you all for your help! I really appreciate your work :)