Wieferich primes and identities for the Euler quotients of $2^n+1$ and $\frac{2^n+1}{3}$

Question

Let $n>1$ be odd integer.

Define the Euler quotient $a(n)=\frac{2^{\varphi(n)}-1 \bmod n^2}{n}$.

Number $n$ with $a(n)=0$ is Wieferich number and if it is prime it is Wieferich prime.

It is open problem if there are only finitely many Wieferich primes and if there are finitely many non-Wieferich primes.

Define $W(n)=\frac{2^n+1}{3}$ and $V(n)=2^n+1$.

If $W(n)$ is prime it is called Wagstaff prime.

Experimentally for $n$ up to $200$ we have the identities:

$$n (-V(n) + a(V(n))) + \varphi(V(n))=0 \qquad (1)$$ $$n (-W(n) + a(W(n))) + 3\varphi(W(n))=0 \qquad (2)$$

Q1 Are the identities true?

We believe that proving the identities will show $a(n)$ can't be zero, so $W(n),V(n)$ are not Wieferich numbers.

It is related to the solved question

sagemath code

def W(n):  return (2**n+1)//3
def V(n):  return 2**n+1
def a(n,B=2):  return lift((Integers(n**2)(B))**(euler_phi(n))-1)/n

for n in range(3,200,2):
    v1=n*(-V(n) + a(V(n))) + euler_phi(V(n))
    v2=n*(-W(n) + a(W(n))) + 3*euler_phi(W(n))
    print(n,v1==0,v2==0) #,a(V(n))/a(W(n)))