Expression 24: "I" Subscript, "s" "I" "t" "h" "P" "r" "i" "m" "e" "F" "a" "c" "t" "o" "r" , Baseline left parenthesis, "n" , "i" , "k" , right parenthesis equals left brace, "N" Subscript, "u" "m" "b" "e" "r" "O" "f" "B" "o" "u" "n" "d" "e" "d" "P" "r" "i" "m" "e" "F" "a" "c" "t" "o" "r" "s" , Baseline left parenthesis, "n" , "k" , right parenthesis greater than or equal to "i" : 1 , 0 , right brace left brace, StartAbsoluteValue, "N" Subscript, "u" "m" "b" "e" "r" "O" "f" "B" "o" "u" "n" "d" "e" "d" "P" "r" "i" "m" "e" "F" "a" "c" "t" "o" "r" "s" , Baseline left parenthesis, "n" , "k" , right parenthesis minus "i" , EndAbsoluteValue less than "T" Subscript, "i" "m" "e" "s" "D" "i" "v" "i" "s" "i" "b" "l" "e" , Baseline left parenthesis, "n" , "k" , right parenthesis : 1 , 0 , right braceIsIthPrimeFactorn,i,k=NumberOfBoundedPrimeFactorsn,k≥i:1,0NumberOfBoundedPrimeFactorsn,k−i<TimesDivisiblen,k:1,0
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What is the prime factorisation of n?
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Expression 26: "P" Subscript, "r" "i" "m" "e" "F" "a" "c" "t" "o" "r" "i" "s" "a" "t" "i" "o" "n" , Baseline left parenthesis, "n" , right parenthesis equals Start sum from "i" Subscript, 6 , Baseline equals left bracket, 1... "N" Subscript, "u" "m" "b" "e" "r" "O" "f" "P" "r" "i" "m" "e" "F" "a" "c" "t" "o" "r" "s" , Baseline left parenthesis, "n" , right parenthesis , right bracket to left bracket, 1... "N" Subscript, "u" "m" "b" "e" "r" "O" "f" "P" "r" "i" "m" "e" "F" "a" "c" "t" "o" "r" "s" , Baseline left parenthesis, "n" , right parenthesis , right bracket , end sum, "P" Subscript, "r" "i" "m" "e" "F" "a" "c" "t" "o" "r" , Baseline left parenthesis, "n" , "i" Subscript, 6 , Baseline , right parenthesisPrimeFactorisationn=1...NumberOfPrimeFactorsn∑i6=1...NumberOfPrimeFactorsnPrimeFactorn,i6
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Visualisation Functions
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Expression 28: "P" equals "P" Subscript, "r" "i" "m" "e" "F" "a" "c" "t" "o" "r" "i" "s" "a" "t" "i" "o" "n" , Baseline left parenthesis, "N" , right parenthesisP=PrimeFactorisationN
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Expression 29: "C" Subscript, "i" "r" "c" "l" "e" "s" , Baseline left parenthesis, "p" , right parenthesis equals Start sum from "i" equals 1 to length left parenthesis, "p" , right parenthesis , end sum, Start product from "j" equals 1 to "i" , end product, "p" left bracket, "j" , right bracketCirclesp=lengthp∑i=1i∏j=1pj
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Expression 30: "A" Subscript, "d" "d" , Baseline left parenthesis, "z" , "w" , right parenthesis equals left parenthesis, "z" . "x" plus "w" . "x" , "z" . "y" plus "w" . "y" , right parenthesisAddz,w=z.x+w.x,z.y+w.y
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Expression 31: "S" Subscript, "c" "a" "l" "e" , Baseline left parenthesis, "k" , "z" , right parenthesis equals left parenthesis, "k" "z" . "x" , "k" "z" . "y" , right parenthesisScalek,z=kz.x,kz.y
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Expression 32: "M" equals "C" Subscript, "i" "r" "c" "l" "e" "s" , Baseline left parenthesis, "P" , right parenthesisM=CirclesP
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Expression 33: "I" left parenthesis, "t" , right parenthesis equals left brace, 0 less than or equal to "t" less than 1 : 1 , 0 , right braceIt=0≤t<1:1,0
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Expression 34: "K" left parenthesis, "i" , right parenthesis equals StartRoot, StartFraction, 1 minus cos left parenthesis, StartNestedFraction, tau NestedOver "i" , EndNestedFraction , right parenthesis Over 2 , EndFraction , EndRootKi=1−cosτi2
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Expression 35: "Q" left parenthesis, "n" , right parenthesis equals Start product from "j" Subscript, 1 , Baseline equals 1 to "n" , end product, "P" left bracket, "j" Subscript, 1 , Baseline , right bracketQn=n∏j1=1Pj1
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Expression 36: "S" left parenthesis, "x" , right parenthesis equals Start product from "j" Subscript, 1 , Baseline equals 0 to "x" , end product, StartFraction, "K" left parenthesis, "P" left bracket, "j" Subscript, 1 , Baseline plus 1 , right bracket , right parenthesis Over 1 plus "K" left parenthesis, "P" left bracket, "j" Subscript, 1 , Baseline plus 1 , right bracket , right parenthesis , EndFractionSx=x∏j1=0KPj1+11+KPj1+1
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Expression 37: "N" Subscript, "t" , Baseline left parenthesis, "n" , "t" , right parenthesis equals ceiling left parenthesis, "n" "t" , right parenthesisNtn,t=ceilnt
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Expression 38: "T" left parenthesis, "n" , "t" , right parenthesis equals mod left parenthesis, "n" "t" , 1 , right parenthesisTn,t=modnt,1
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Expression 39: "O" left parenthesis, "n" , "t" , right parenthesis equals left parenthesis, StartFraction, cos left parenthesis, tau StartNestedFraction, "N" Subscript, "t" , Baseline left parenthesis, "n" , "t" , right parenthesis NestedOver "n" , EndNestedFraction , right parenthesis Over 1 plus "K" left parenthesis, "n" , right parenthesis , EndFraction plus 10 to the negative 5th power "T" left parenthesis, "n" , "t" , right parenthesis , StartFraction, sin left parenthesis, tau StartNestedFraction, "N" Subscript, "t" , Baseline left parenthesis, "n" , "t" , right parenthesis NestedOver "n" , EndNestedFraction , right parenthesis Over 1 plus "K" left parenthesis, "n" , right parenthesis , EndFraction , right parenthesisOn,t=cosτNtn,tn1+Kn+10−5Tn,t,sinτNtn,tn1+Kn
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Expression 40: "C" left parenthesis, "n" , "p" , "s" , "t" , right parenthesis equals left parenthesis, "p" . "x" plus "s" StartFraction, cos left parenthesis, tau StartNestedFraction, "N" Subscript, "t" , Baseline left parenthesis, "n" , "t" , right parenthesis NestedOver "n" , EndNestedFraction , right parenthesis plus "K" left parenthesis, "n" , right parenthesis cos left parenthesis, tau "T" left parenthesis, "n" , "t" , right parenthesis , right parenthesis Over 1 plus "K" left parenthesis, "n" , right parenthesis , EndFraction , "p" . "y" plus "s" StartFraction, sin left parenthesis, tau StartNestedFraction, "N" Subscript, "t" , Baseline left parenthesis, "n" , "t" , right parenthesis NestedOver "n" , EndNestedFraction , right parenthesis plus "K" left parenthesis, "n" , right parenthesis sin left parenthesis, tau "T" left parenthesis, "n" , "t" , right parenthesis , right parenthesis Over 1 plus "K" left parenthesis, "n" , right parenthesis , EndFraction , right parenthesisCn,p,s,t=p.x+scosτNtn,tn+KncosτTn,t1+Kn,p.y+ssinτNtn,tn+KnsinτTn,t1+Kn
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Visualisation
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Expression 42: left parenthesis, cos tau "t" , sin tau "t" , right parenthesiscosτt,sinτt
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Expression 43: left parenthesis, Start sum from "k" Subscript, 1 , Baseline equals 0 to length left parenthesis, "P" , right parenthesis minus 1, end sum, "I" left parenthesis, "t" minus "k" Subscript, 1 , Baseline , right parenthesis left parenthesis, "S" left parenthesis, "k" Subscript, 1 , Baseline minus 1 , right parenthesis StartFraction, cos left parenthesis, tau StartNestedFraction, "N" Subscript, "t" , Baseline left parenthesis, "P" left bracket, "k" Subscript, 1 , Baseline plus 1 , right bracket , "T" left parenthesis, "Q" left parenthesis, "k" Subscript, 1 , Baseline , right parenthesis , "t" , right parenthesis , right parenthesis NestedOver "P" left bracket, "k" Subscript, 1 , Baseline plus 1 , right bracket , EndNestedFraction , right parenthesis plus "K" left parenthesis, "P" left bracket, "k" Subscript, 1 , Baseline plus 1 , right bracket , right parenthesis cos left parenthesis, tau "T" left parenthesis, "P" left bracket, "k" Subscript, 1 , Baseline plus 1 , right bracket , "T" left parenthesis, "Q" left parenthesis, "k" Subscript, 1 , Baseline , right parenthesis , "t" , right parenthesis , right parenthesis , right parenthesis Over 1 plus "K" left parenthesis, "P" left bracket, "k" Subscript, 1 , Baseline plus 1 , right bracket , right parenthesis , EndFraction plus Start sum from "k" Subscript, 3 , Baseline equals 1 to "k" Subscript, 1 , Baseline , end sum, "S" left parenthesis, "k" Subscript, 3 , Baseline minus 2 , right parenthesis StartFraction, cos left parenthesis, tau StartNestedFraction, "N" Subscript, "t" , Baseline left parenthesis, "P" left bracket, "k" Subscript, 3 , Baseline , right bracket , "T" left parenthesis, "Q" left parenthesis, "k" Subscript, 3 , Baseline minus 1 , right parenthesis , "t" , right parenthesis , right parenthesis NestedOver "P" left bracket, "k" Subscript, 3 , Baseline , right bracket , EndNestedFraction , right parenthesis Over left parenthesis, 1 plus "K" left parenthesis, "P" left bracket, "k" Subscript, 3 , Baseline , right bracket , right parenthesis , right parenthesis , EndFraction , right parenthesis , Start sum from "k" Subscript, 1 , Baseline equals 0 to length left parenthesis, "P" , right parenthesis minus 1, end sum, "I" left parenthesis, "t" minus "k" Subscript, 1 , Baseline , right parenthesis left parenthesis, "S" left parenthesis, "k" Subscript, 1 , Baseline minus 1 , right parenthesis StartFraction, sin left parenthesis, tau StartNestedFraction, "N" Subscript, "t" , Baseline left parenthesis, "P" left bracket, "k" Subscript, 1 , Baseline plus 1 , right bracket , "T" left parenthesis, "Q" left parenthesis, "k" Subscript, 1 , Baseline , right parenthesis , "t" , right parenthesis , right parenthesis NestedOver "P" left bracket, "k" Subscript, 1 , Baseline plus 1 , right bracket , EndNestedFraction , right parenthesis plus "K" left parenthesis, "P" left bracket, "k" Subscript, 1 , Baseline plus 1 , right bracket , right parenthesis sin left parenthesis, tau "T" left parenthesis, "P" left bracket, "k" Subscript, 1 , Baseline plus 1 , right bracket , "T" left parenthesis, "Q" left parenthesis, "k" Subscript, 1 , Baseline , right parenthesis , "t" , right parenthesis , right parenthesis , right parenthesis Over 1 plus "K" left parenthesis, "P" left bracket, "k" Subscript, 1 , Baseline plus 1 , right bracket , right parenthesis , EndFraction plus Start sum from "k" Subscript, 3 , Baseline equals 1 to "k" Subscript, 1 , Baseline , end sum, "S" left parenthesis, "k" Subscript, 3 , Baseline minus 2 , right parenthesis StartFraction, sin left parenthesis, tau StartNestedFraction, "N" Subscript, "t" , Baseline left parenthesis, "P" left bracket, "k" Subscript, 3 , Baseline , right bracket , "T" left parenthesis, "Q" left parenthesis, "k" Subscript, 3 , Baseline minus 1 , right parenthesis , "t" , right parenthesis , right parenthesis NestedOver "P" left bracket, "k" Subscript, 3 , Baseline , right bracket , EndNestedFraction , right parenthesis Over left parenthesis, 1 plus "K" left parenthesis, "P" left bracket, "k" Subscript, 3 , Baseline , right bracket , right parenthesis , right parenthesis , EndFraction , right parenthesis , right parenthesislengthP−1∑k1=0It−k1Sk1−1cosτNtPk1+1,TQk1,tPk1+1+KPk1+1cosτTPk1+1,TQk1,t1+KPk1+1+k1∑k3=1Sk3−2cosτNtPk3,TQk3−1,tPk31+KPk3,lengthP−1∑k1=0It−k1Sk1−1sinτNtPk1+1,TQk1,tPk1+1+KPk1+1sinτTPk1+1,TQk1,t1+KPk1+1+k1∑k3=1Sk3−2sinτNtPk3,TQk3−1,tPk31+KPk3
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Expression 44: left parenthesis, 0 , 0 , right parenthesis0,0
