Divisor

Author

Eric W. Weisstein
December 21, 2009

This notebook downloaded from http://mathworld.wolfram.com/notebooks/NumberTheory/Divisor.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/Divisor.html.

©2009 Wolfram Research, Inc. except for portions noted otherwise

List of Divisors

Table[{n,Divisors[n]},{n,15}]//ColumnForm

Flatten[Table[Divisors[n],{n,15}]]

PrimeOmega[n]

Series[PrimeOmega[n],{n,0,1}]

Series[PrimeOmega[n],{n,∞,1}]

D[PrimeOmega[n],n]

FunctionExpand[PrimeOmega[n]]

PrimeNu[n]

Series[PrimeNu[n],{n,0,1}]

Series[PrimeNu[n],{n,∞,1}]

D[PrimeNu[n],n]

FunctionExpand[PrimeNu[n]]

Smallest Number having divisor n

d=First/@Split[Sort[{DivisorSigma[0,#],#}&/@Range[10^6]],#1[[1]]==#2[[1]]&];//Timing

d=First/@Split[Sort[{DivisorSigma[0,#],#}&/@Range[10^7]],#1[[1]]==#2[[1]]&];//Timing

p=Position[Subtract@@@Partition[First/@d,2,1],_?(#≠-1&),1,1][[1,1]]

l=Last/@Take[d,p]

l={1,2,4,6,16,12,64,24,36,48,1024,60,4096,192,144,120,65536,180,262144,240,576,3072,4194304,360,1296,12288,900,960};

Fontené's Theorem

FactorInteger/@l

munu=If[Length[#]==1,{1+#[[1,-1]]},(1+Last/@Take[#,-2])]&/@
FactorInteger/@Rest[l]

Map[PrimeQ,munu,{1}]

Plot

<<Utilities`Plot`

Show[PixelPerfect[NaryPlot[Last/@Take[d,p]],BlockSize->3]]

Identities

FullSimplify[PrimeOmega[n]-PrimeNu[n],n∈Primes]

Table[-PrimeNu[n]+PrimeOmega[n]/.n->Prime[k],{k,1000}]//Union

Sums

DivisorSum[n,1&]==(DivisorSigma[0, n])

DivisorSum[n,1&]

DivisorSum[n,((#1)^k)&]==(DivisorSigma[k, n])

DivisorSum[n,#^k&]

DivisorSum[n,((#1)^(1-s))&]==(n^(1-s)) (DivisorSigma[s-1, n])==(DivisorSigma[1-s, n])

DivisorSum[n,((#)^(1-s))&]

Subtract@@@Table[{DivisorSum[n,((#)^(1-s))&],(n^(1-s))DivisorSigma[s-1,n]},{n,100}]//Simplify//Union

Subtract@@@Table[{DivisorSum[n,((#)^(1-s))&],DivisorSigma[1-s,n]},{n,100}]//Simplify//Union

{DivisorSum[n,((#)^(1-s))&],(n^(1-s))DivisorSigma[s-1,n],DivisorSigma[1-s,n]}/.{n->10,s->0}

{DivisorSum[n,((#)^(1-s))&],(n^(1-s))DivisorSigma[s-1,n],DivisorSigma[1-s,n]}/.{n->10,s->-2}

{DivisorSum[n,((#)^(1-s))&],(n^(1-s))DivisorSigma[s-1,n],DivisorSigma[1-s,n]}/.{n->10,s->1}

{DivisorSum[n,((#)^(1-s))&],(n^(1-s))DivisorSigma[s-1,n],DivisorSigma[1-s,n]}/.{n->10,s->-2}

4DivisorSum[n,(((-1))^((#1-1)/2))&,OddQ]==(SquaresR[2, n])

4DivisorSum[n,(((-1))^((#-1)/2))&,OddQ]

4 (DivisorSum[n,1&,#1 mod 4==1&]-DivisorSum[n,1&,#1 mod 4==3&])==(SquaresR[2, n])

4(DivisorSum[n,1&,Mod[#,4]==1&]-DivisorSum[n,1&,Mod[#,4]==3&])

FullSimplify[%,n>0&&n∈Integers]

4DivisorSum[n,sin(((π #1)/2))&]==(SquaresR[2, n])

4DivisorSum[n,Sin[((π #)/2)]&]

8 DivisorSum[n,#1&,#1 mod 4≠0&]==(SquaresR[4, n])

8DivisorSum[n,#&,Mod[#,4]≠0&]

16DivisorSum[n,((#1)^3) (((-1))^(#1+n))&]==(SquaresR[8, n])

16DivisorSum[n,(((-1))^(n+#))((#)^3)&]

(∑_n^∞)(q^n) DivisorSum[n,(KroneckerSymbol[-6, #1])&]+1==(((QPochhammer1[(q^2)]) (QPochhammer1[(q^3)]) (QPochhammer1[(q^8)]) (QPochhammer1[(q^12)]))/((QPochhammer1[q]) (QPochhammer1[(q^24)])))

Fine's Equations

1+Sum[DivisorSum[n,KroneckerSymbol[-6,#]&](q^n),{n,∞}]//InputForm

Subtract@@@Table[{1+Sum[DivisorSum[n,KroneckerSymbol[-6,#]&](q^n),{n,10^3}],(QPochhammer[(q^2)]QPochhammer[(q^3)]QPochhammer[(q^8)]QPochhammer[(q^12)])/(QPochhammer[q]QPochhammer[(q^24)])}/.q->RandomReal[{-1,1},WorkingPrecision->20],{50}]//Chop//Union

DivisorSum[n,(MoebiusMu[#1])&]==(Subscript[δ, 1,n])

DivisorSum[n,MoebiusMu]

DivisorSum[n,MoebiusMu[#]&]

DivisorSum[n,(MoebiusMu[#1]) (DivisorSigma[0, (n/(#1))])&]==1

DivisorSum[n,MoebiusMu[#]DivisorSigma[0,(n/(#))]&]

DivisorSum[n,(EulerPhi[#1])&]==n

DivisorSum[n,EulerPhi[#]&]

DivisorSum[n,EulerPhi]

DivisorSum[n,#1 (MoebiusMu[(n/(#1))])&]==(EulerPhi[n])

DivisorSum[n,# MoebiusMu[(n/(#))]&]

((DivisorSum[n,(MoebiusMu[#1]) (2^(n/#))&])/n)==i(n)

Twin primes constant

((DivisorSum[n,MoebiusMu[#](2^(n/#))&])/n)

Table[((DivisorSum[n,MoebiusMu[#](2^(n/#))&])/n),{n,10}]

DivisorSum[n,1&,PrimeQ[#1]&]==PrimeNu[n]

V7

DivisorSum[n,1&,PrimeQ]

DivisorSum[n,1&,PrimeQ[#]&]

DivisorSum[n,1&,Element[#,Primes]&]

V8

DivisorSum[n,1&,PrimeQ]

DivisorSum[n,1&,PrimeQ[#]&]

DivisorSum[n,1&,Element[#,Primes]&]

DivisorSum[n,((#1)^k)&,EvenQ[#1]&]==If[EvenQ[n],(2^k) (DivisorSigma[k, (n/2)]),0]

DivisorSum[n,#^2&,EvenQ]

DivisorSum[n,#^2&,EvenQ[#]&]

Table[DivisorSum[n,#^k&,EvenQ[#]&],{k,4}]

Subtract@@@Flatten[Table[{DivisorSum[n,((#)^k)&,EvenQ[#]&],If[EvenQ[n],(2^k)DivisorSigma[k,n/2],0]},{k,5},{n,100}],1]//Union

DivisorSum[n,(1/(#1))&]==((DivisorSigma[1, n])/n)

DivisorSum[n,1/#&]

exp(DivisorSum[n,ln(#1)&])==(n^((DivisorSigma[0, n])/2))

Exp[DivisorSum[n,Log[#]&]]

DivisorSum[n,ln(#1) (MoebiusMu[(n/(#1))])&]==MangoldtLambda[n]

DivisorSum[n,MoebiusMu[(n/(#))]Log[#]&]

DivisorSum[n,ln(#1) (MoebiusMu[#1])&]==-MangoldtLambda[n]

DivisorSum[n,MoebiusMu[#]Log[#]&]

DivisorSum[n,(MoebiusMu[(n/(#1))]) MangoldtLambda[#1]&]==ln(n) (-(MoebiusMu[n]))

DivisorSum[n,MoebiusMu[(n/(#))]MangoldtLambda[#]&]

Subtract@@@Table[{DivisorSum[n,MoebiusMu[(n/(#))]MangoldtLambda[#]&],-MoebiusMu[n]Log[n]},{n,1000}]//Simplify//Union

DivisorSum[n,MangoldtLambda[#1]&]==ln(n)

DivisorSum[n,MangoldtLambda]

DivisorSum[n,MangoldtLambda[#]&]

DivisorSum[n, Log[XXX] &, PrimePowerQ[#] &]

DivisorSum[n,Log[#]&,PrimePowerQ]

DivisorSum[n,Log[#]&,PrimePowerQ[#]&]

Table[{DivisorSum[n,Log[p&],PrimePowerQ[p^a=#]&],Log[n]},{n,10}]