User:Enrique Pérez Herrero/EulerPhi-Divisor

Diophantine Equations including Divisor Function

Sequences in OEIS

Diophantine Equations Related to Divisor Function

Equation	Sequence Id
$\displaystyle \phi (n)=\tau (n)$	A020488
$\displaystyle \phi (n)=2\cdot \tau (n)$	A062516
$\displaystyle \phi (n)=3\cdot \tau (n)$	A063469
$\displaystyle \phi (n)=4\cdot \tau (n)$	A063470
$\displaystyle \phi (n)=k\cdot \tau (n)$	A112954, A112955, A175667
$\displaystyle \phi (n)={\tau (n)}^{2}$	A068560
$\displaystyle \phi (n)={\tau (n)}^{3}$	A068559
$\displaystyle \phi (n)={\tau (n)}^{4}$	A114063

NOTES ON $\displaystyle \phi (n)=k\cdot \tau (n)$:

The Number of Solutions is Finite

• Upper bound for $\displaystyle \tau (n)$:

$\displaystyle \tau (n)<n^{\frac {2}{3}};$ $\displaystyle (n>11)$

• Lower bound for $\displaystyle \phi (n)$:

$\displaystyle \phi (n)>{\frac {n}{e^{\gamma }\cdot \log(\log(n))+{\frac {3}{\log(\log(n))}}}}$

Mathematica Code:

tauUpperBound[n_] := n^(2/3);
phiLowerBound[n_] := n/(Exp[EulerGamma]*Log[Log[n]] + 3/(Log[Log[n]]));
SolutionsBound[k_Integer] := Part[Floor[n/.NSolve[Abs[k*tauUpperBound[n]-phiLowerBound[n]] == 0, n, Reals]], 2]

(* Code for ploting phi(n)=tau(n) with bounds*)
g1 = ListPlot[Table[EulerPhi[n], {n, 1, 200}], Joined -> True];
g2 = ListPlot[Table[DivisorSigma[0, n], {n, 1, 200}], Joined -> True, 
 PlotStyle -> Red];
g3 = Plot[{tauUpperBound[n], phiLowerBound[n]}, {n, 5, 200}, 
 PlotStyle -> {Red, Blue}];
Show[g1, g2, g3]

Bound for solutions

k	Sequence	Bound	A175667(k)	A112955(k)	$S_{k}$
1	A020488	103	1	30	{1, 3, 8, 10, 18, 24, 30}
2	A062516	996	5	120	{5, 9, 15, 28, 40, 72, 84, 90, 120}
3	A063469	3746	7	210	{7, 21, 26, 56, 70, 78, 108, 126, 168, 210}
4	A063470	9525	34	420	{34, 45, 52, 102, 140, 156, 252, 360, 420}
5	-	19568	11	330	{11, 33, 88, 110, 198, 264, 330}
6	-	35158	13	840	{13, 35, 39, 63, 76, 104, 105, 130, 228, 234, 280, 312, 390, 504, 540, 630, 840}

Properties:

• 1) If p=2k+1 is a prime then n=p is a solution of the equation and then A112954(k)>0 and also A112955(k)>0.

• 2) If p=2k+1 is a prime distinct than 3, then n=3*p is also a solution.

• 3) If p=2k+1 is a prime distinct than 5, then n=10*p is also a solution.

• 4) If p=2k+1 is a prime >5, then n=30*p is also a solution.

• 5) If p=2k+1 is a prime and k>1 there are no solutions for n<p.

• 6) If $\phi (a)=k_{1}\cdot \tau (a)$, $\phi (b)=k_{2}\cdot \tau (b)$, and $a\perp b$, then: $\phi (a\cdot b)=k_{1}\cdot k_{2}\cdot \tau (a\cdot b)$.

• 7) Supose that $a\in S_{k}$, is a solution to the equation $\phi (a)=k\cdot \tau (a)$, with the set of solutions for the case $k=1$:

$S_{1}=\{1,3,8,10,18,24,30\}$ (Sequence A020488), it is posible to construct more solutions for the case $k$, if $b\in S_{1}$, and $a\perp b$ like $\phi (n)$ and $\tau (n)$ are multiplicative arithmetical functions then: $\phi (a\cdot b)=k\cdot \tau (a\cdot b)$, so $a\cdot b$ is also a solution to the equation.: $a\cdot b\in S_{k}$.

(* This code displays new solutions from an already known one, using sequence A020488*)
S[n_]:=Select[{1, 3, 8, 10, 18, 24, 30}, CoprimeQ[#, n] &]*n

• 8) If p=2k+1 is a prime >5 then $\phi (n)=k\cdot \tau (n)$, has at least 7 solutions: A112954(k) $\geq$ 7

• 9) Conjecture:(False!!!) The equation $\displaystyle \phi (n)=k\cdot \tau (n)$, has no solution if $\displaystyle k>2$ is in the sequence: A119480, Numbers n such that the Bernoulli number B_{4n} has denominator 30.

if this conjecture holds then: A112954 (A119480(k)) =A112955(A119480(k)=0 ; (k>2)

Counterexample found: $\phi (1132)=94\cdot \tau (1132)$

• 10) Conjecture:: if k>3 is a Sophie-Germain prime (A005384) then the problem has just 7 solutions, given by 7)

• 11) If p=4k+1 is a prime then 2p is a solution for k.

• 12) If p=3k+1 is prime then 4p is a solution for k.

References:

• Jud McCranie, Comment to A020488
• J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. Volume 6, Issue 1 (1962), 64-94.