Algebra of quaternions and octonions involving higher order Mersenne numbers

In this article, we investigate the algebra of quaternions and octonions with components as the higher order Mersenne numbers. We examine the inter-relations of these newly introduced sequences with the Mersenne numbers and their quaternions and octonions counterparts. Moreover, we obtain the Binet type formula, recurrence relation, Catalan's identity, d'Ocagne's identity, Vajda's identity, Honsberger's identity, generating functions, summations formulas, etc. of them.