Density conjecture for weak primary pseudoperfect numbers

Let QQ and QQ' be weak primary pseudoperfect numbers. Let NQ\mathfrak{N}_Q denote the set of positive integers associated with QQ in the paper, and let nQ\mathfrak{n}_Q be its least element when NQ\mathfrak{N}_Q is non-empty; write δ(NQ)\delta(\mathfrak{N}_Q) for its asymptotic density.

Density conjecture. If NQ\mathfrak{N}_Q and NQ\mathfrak{N}_{Q'} are non-empty and nQ<nQ\mathfrak{n}_Q<\mathfrak{n}_{Q'}, then

δ(NQ)>δ(NQ).\delta(\mathfrak{N}_Q)>\delta(\mathfrak{N}_{Q'}).

The prediction is motivated by the computed examples in the paper, but the source does not establish it. Its general validity remains open.

Sources & referencesView supporting material

Primary source

José María Grau, Antonio M. Oller-Marcén and Jonathan Sondow, “On the congruence 1^m + 2^m + + m^m n m with n | m”, arXiv:1309.7941 (2014).

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