  Let $\mathbb{Y}$ be either an Orlicz sequence space or a Marcinkiewicz
sequence space. We take advantage of the recent advances in the theory of
factorization of the identity carried on in [R. Lechner, Subsymmetric weak*
Schauder bases and factorization of the identity, arXiv:1804.01372 [math.FA]]
to provide conditions on $\mathbb{Y}$ that ensure that, for any $1\le
p\le\infty$, the infinite direct sum of $\mathbb{Y}$ in the sense of $\ell_p$
is a primary Banach space, enlarging this way the list of Banach spaces that
are known to be primary.
