/LMAR=0/XLINE=3/FONT#0=BASB30/FONT#1=BDR66/FONT#3=BDI40/FONT#6=BDR40/FONT#7=BDR25/FONT#8=GRFX35/TMAR=50/PMAR=2100/BMAR=50





␈↓ d␈↓↓␈↓ ∧ ␈↓&Facets of a Concept␈↓)αβ


␈↓ d␈↓↓Characterizations
␈↓ d␈↓↓        ␈↓εName(s)␈↓↓
␈↓ d␈↓↓        ␈↓εDefinitions ␈↓↓
␈↓ d␈↓↓        ␈↓εAlgorithms ␈↓↓
␈↓ d␈↓↓        ␈↓εDomain/range␈↓↓
␈↓ d␈↓↓        ␈↓εIntuitions: abstract representations␈↓↓

␈↓ d␈↓↓Ties to other concepts
␈↓ d␈↓↓        ␈↓εSpecializations␈↓↓
␈↓ d␈↓↓        ␈↓εGeneralizations␈↓↓
␈↓ d␈↓↓        ␈↓εExamples␈↓↓
␈↓ d␈↓↓        ␈↓εOperations one can do to this concept␈↓↓
␈↓ d␈↓↓        ␈↓εConjectures/theorems involving this concept␈↓↓
␈↓ d␈↓↓        ␈↓εAnalogies␈↓↓

␈↓ d␈↓↓Heuristics
␈↓ d␈↓↓        ␈↓εWorth: Why this concept is worth naming␈↓↓
␈↓ d␈↓↓        ␈↓εInterest: When an instance of it is (un)interesting␈↓↓
␈↓ d␈↓↓        ␈↓εFillin: Hints for filling in parts of instances␈↓↓
␈↓ d␈↓↓        ␈↓εSuggest new activities for AM to consider␈↓↓
␈↓ d␈↓↓        ␈↓εCheck: things to watch out for␈↓↓





␈↓ d␈↓↓␈↓ βs␈↓&Facets of  "COMPOSE"␈↓)αβ


␈↓ d␈↓↓Characterizations
␈↓ d␈↓↓        ␈↓εName(s): Compose ␈↓↓
␈↓ d␈↓↓        ␈↓εDefinitions: recursive, opaque, wffs␈↓↓
␈↓ d␈↓↓        ␈↓εAlgorithms: opaque, transparent, destructive ␈↓↓
␈↓ d␈↓↓        ␈↓εDomain/range: Relations x Relations → Relations␈↓↓
␈↓ d␈↓↓        ␈↓εIntuitions: refiring arrows, time sequence␈↓↓

␈↓ d␈↓↓Ties to other concepts
␈↓ d␈↓↓        ␈↓εSpecializations: Compose f with itself␈↓↓
␈↓ d␈↓↓        ␈↓εGeneralizations: Relation␈↓↓
␈↓ d␈↓↓        ␈↓εExamples: (Intersect, Complement) → Set-difference␈↓↓
␈↓ d␈↓↓        ␈↓εConjec: (AoB)oC ≡ Ao(BoC)␈↓↓
␈↓ d␈↓↓        ␈↓εAnalogies: multiplying two matrices␈↓↓

␈↓ d␈↓↓Heuristics
␈↓ d␈↓↓        ␈↓εWorth: Primitive. Creates new active Concepts␈↓↓
␈↓ d␈↓↓        ␈↓εInterest: Domain=Range; both args are interesting␈↓↓
␈↓ d␈↓↓        ␈↓εFillin: D/R are Domain(arg1) and Range(arg2)␈↓↓
␈↓ d␈↓↓        ␈↓εSugg: Check AoB for properties which A or B have␈↓↓
␈↓ d␈↓↓        ␈↓εCheck: Domain(arg2) should intersect Range(arg1)␈↓↓





␈↓ d␈↓↓␈↓ ∧:␈↓&Control Structure␈↓)αβ



␈↓ d␈↓εTHINGS WORTH DOING


␈↓ d␈↓ε(Fill in exs of Primes)

␈↓ d␈↓ε(Improve algs for Compose)

␈↓ d␈↓ε(Generalize Defn of Equality)









␈↓ d␈↓ε␈↓ ¬,Select 1 activity




␈↓ d␈↓εExecute this plan

␈↓ d␈↓ε␈↓ λ≥Assemble relevant heuristics





␈↓ d␈↓↓␈↓ βr␈↓&Graph of Development␈↓)αβ





␈↓ d␈↓εBags␈↓ ∧4Equality␈↓ εtCross-product



␈↓ d␈↓ε␈↓ ∧4Numbers␈↓ εt␈↓ λ∀Projection



␈↓ d␈↓ε␈↓ ∧4Multiplication



␈↓ d␈↓εExponentiation␈↓ ∧4Divisors



␈↓ d␈↓εHyper-exponentiation␈↓ ∧4Max-Divis␈↓ εtPrimes





␈↓ d␈↓↓␈↓ ∧⊂␈↓&Complete the Square␈↓)αβ


␈↓ d␈↓↓␈↓ ¬T␈↓εcount␈↓↓
␈↓ d␈↓↓Pairs of Bags---------------------␈↓ π]→ Pairs of Numbers
␈↓ d␈↓↓    |␈↓ ¬T␈↓ λT␈↓ 
∧|
␈↓ d␈↓↓    |␈↓ ¬T␈↓ λT␈↓ 
∧
␈↓ d␈↓↓    |␈↓ ¬T␈↓ λT␈↓ 
∧|
␈↓ d␈↓↓    |␈↓ ¬T␈↓ λT␈↓ 
∧
␈↓ d␈↓↓    |␈↓ ¬T␈↓ λT␈↓ 
∧|
␈↓ d␈↓↓    |␈↓ ¬T␈↓ λT␈↓ 
∧
␈↓ d␈↓↓    | ␈↓εcross-product␈↓↓␈↓ ¬T␈↓ λT␈↓ 
∧|  ␈↓ε(?)␈↓↓
␈↓ d␈↓↓    |␈↓ ¬T␈↓ λT␈↓ 
∧
␈↓ d␈↓↓    |␈↓ ¬T␈↓ λT␈↓ 
∧|
␈↓ d␈↓↓    |␈↓ ¬T␈↓ λT␈↓ 
∧
␈↓ d␈↓↓    |␈↓ ¬T␈↓ λT␈↓ 
∧|
␈↓ d␈↓↓    |␈↓ ¬T␈↓ λT␈↓ 
∧
␈↓ d␈↓↓    ↓␈↓ ¬T␈↓εcount␈↓↓␈↓ λT␈↓ 
∧↓
␈↓ d␈↓↓Bags--------------------␈↓ ¬T----------------␈↓ λT→ Numbers



␈↓ d␈↓ε?(x,y) = Count ( Cross-product ( Count␈↓#
-␈↓#␈↓#
1␈↓#(x), Count␈↓#
-␈↓#␈↓#
1␈↓#(y) ) ).





␈↓ d␈↓↓␈↓ β∀␈↓&Maximally Divisible Numbers␈↓)αβ


␈↓ d␈↓↓Max-divis(N) iff (∀m<n) d(m) < d(n)

␈↓ d␈↓↓     if  N  =  2␈↓#
a␈↓#␈↓#
⊗␈↓#713␈↓#
a␈↓#␈↓ε2␈↓↓5␈↓#
a␈↓#␈↓ε3␈↓↓...p␈↓ε␈↓#vk␈↓#␈↓↓␈↓#
a␈↓#␈↓εk␈↓↓

␈↓ d␈↓↓where   p␈↓#vi␈↓#  is  the  i␈↓#
t␈↓#␈↓#
h␈↓#  prime,

␈↓ d␈↓↓and   (a␈↓#vi␈↓# + 1) / (a␈↓#vj␈↓# + 1)   "="   log(p␈↓#vj␈↓# ) / log(p␈↓#vi␈↓#)

␈↓ d␈↓↓**************************************************

␈↓ d␈↓↓For example:   n could be

␈↓ d␈↓↓2␈↓ε␈↓#
8␈↓#␈↓↓3␈↓ε␈↓#
5␈↓#␈↓↓5␈↓ε␈↓#
3␈↓#␈↓↓7␈↓ε␈↓#
2␈↓#␈↓↓11␈↓ε␈↓#
2␈↓#␈↓↓13␈↓ε␈↓#
1␈↓#␈↓↓17␈↓ε␈↓#
1␈↓#␈↓↓19␈↓ε␈↓#
1␈↓#␈↓↓23␈↓ε␈↓#
1␈↓#␈↓↓29␈↓ε␈↓#
1␈↓#␈↓↓31␈↓ε␈↓#
1␈↓#␈↓↓37␈↓ε␈↓#
1␈↓#␈↓↓41␈↓ε␈↓#
1␈↓#␈↓↓43␈↓ε␈↓#
1␈↓#␈↓↓47␈↓ε␈↓#
1␈↓#␈↓↓53␈↓ε␈↓#
1␈↓#␈↓↓
␈↓ d␈↓↓        (which equals 25,608,675,584).

␈↓ d␈↓↓(a␈↓#vi␈↓# + 1)'s  are  (9 6 4 3 3 2 2 2 2 2 2 2 2 2 2 2)

␈↓ d␈↓↓n has 3,981,312 divisors.


␈↓ d␈↓↓AM  Conjecture says that
␈↓ d␈↓↓n is the smallest integer with that many divisors.