In what follows I want to prove the following claim.
Claim. let $x\in\bigcap_{p>1}\ell^p(\mathbb N)$. Then, $$ \int_1^2|x|_pdp<\infty \iff \int_1^2|x|^p_pdp<\infty \iff \sum_{j\in\mathbb N} \Phi(x_j)<\infty, $$ where $$\Phi(t)=\frac{|t|(|t|-1)}{\ln|t|},$$ (extended by continuity to $\Phi(0)=0$).
Remarks.
- The function $\Phi$ has a nice graph (it is convex, the derivative is $\Phi'(t)=\frac{t}{(\ln t)^2}$).
- The hypothesis $x\in\bigcap_{p>1}\ell^p(\mathbb N)$ is just for context, one simply needs $x\in \mathbb R^{\mathbb N}$ (with the convention that the $\ell^p-$norm of a sequence not belonging to $\ell^p$ is infinite).
- The proof is immediate for the zero sequence, so I will assume that $x\not\equiv 0$.
Lemma 1. Let $f:(0,1]\to(0,+\infty)$ a continuous non-increasing function on $[0,1]$. Then, $$ \int_0^1 f(x)dx<\infty \implies \sup_{x\in(0,1]} xf(x)<+\infty. $$
Proof. (idea; this works only for $C^1$ functions) One has $$\int_0^1 f(x)dx=-\int_0^1 xf'(x)dx.$$ Then, $f(x)=-\int_x^1f'(y)dy\leq \frac{1}{x}\int_0^1yf'(y)dy.$
Lemma 2. Let $f:(0,1]\to(0,+\infty)$ a continuous non-increasing function on $[0,1]$ s.t. $|f(x)|\lesssim 1/x$. Then, $$ \int_0^1 f(x)dx<\infty\iff \int_0^1 |f(x)|^{x+1}dx<\infty.$$
Proof. It comes from the fact that $$ \lim_{x\to 0^+} \frac{|f(x)|^{x+1}}{f(x)}=\lim_{x\to 0^+} |f(x)|^x=1,$$ where the last limit follows by monotonicity as $\lim_{x\to 0^+} 1/x^x=1$ and $\lim_{x\to 0^+} c^x=1$, $c>0$.
Proposition. The first equivalence of the claim holds.
Proof. The direction '$\implies$' follows combining the two Lemmas. The other direction follows by monotonicity.
The remaining part of the claim follows by Fubini's theorem: $$ \int_1^2|x|^p_pdp=\int_1^2\sum_{j\in\mathbb N}|x_j|^pdp=\sum_{j\in\mathbb N}\int_1^2|x_j|^pdp= $$ $$ =\sum_{j\in\mathbb N}\frac{|x_j|^2-|x_j|}{\ln|x_j|}. $$
The claim (together with the observations from other answers) tells that the completion of $\ell^1$ under this norm is the space of sequences such that the above series is finite. I would then want to say that the space is isomorphic to the Orlicz space $\frac{\ell^2-|\ell|}{\ln|\ell|}$, but I know nothing about Orlicz spaces, so maybe someone else can say something about it.