In "THE SEQUENCE SPACES $l(p_\nu)$ AND $m(p_\nu)$" by S. Simons there is the following theorem about sequence spaces of $l_p$-type with varying exponents which is attributed (without a more precise reference) to H.T. Croft and Conway:

For a sequence $p_\nu$ of positive numbers, $l(p_\nu)$ denotes the space of sequences $(a_n)$ such that $\sum_\nu |a_\nu|^{p_\nu}$ is finite and $l_1$ is the usual space of absolutely summable sequences.

Theorem: We suppose that $0 < p_\nu \leq 1$ for all $\nu$, and write $\pi_\nu$ for the conjugate index of $p_\nu$, i.e. $(l/p_\nu) + > (1/\pi_\nu) = 1$, giving $\pi_\nu$ the value $-\infty$ when $p_\nu = > 1$. Then the following are equivalent:

1. $l(p_\nu) = l_1$
2. $\sum_\nu N^{\pi_\nu} < \infty$ for some integer $N > 1$.