Suppose n = d_1 + ... + d_k where the d_i are distinct proper divisors of n but this is not true for any proper divisor of n. Must the sum of the reciprocals of all such n converge? Similarly, the same question can be asked for those n which do not have distinct sums of sets of divisors (but any proper divisor of n does).

Erdős Problem #469 — Suppose n = d_1 + ... + d_k where the d_i are distinct proper divisors of n but this is not true for any proper divisor of n. Must the sum of the reciprocals of all such n converge?

The problem asks whether the reciprocals of the primitive pseudoperfect numbers form a convergent series, with an analogous question for numbers lacking distinct divisor-sum representations while every proper divisor has one. Benkoski and Erdős posed these questions in 1974.

Known results

The catalogue records no published partial or complete result on convergence. A discussion gives a sufficient condition excluding some numbers from the primitive class, but no asymptotic bound for the remainder.

July 2026 claimed solution

Zach Lewis announced an unconditional Lean 4 proof of convergence, based on prime-extension transitions, candidate and forced-prime trees, and weighted control of primitive nondeficient roots. The announcement is self-reported, and no independently verified proof artifact or confirmation was found.

Current status (as of August 2026): Convergence is not independently verified; a complete solution is claimed, but the problem remains open in the verified mathematical record.

Erdős Problem #469 — Suppose $n = d_1 + ... + d_k$ where the $d_i$ are distinct proper divisors of $n$ but this is not true for any proper divisor of $n$ . Must the sum of the reciprocals of all such $n$ converge?

Primary source

Additional references

Progress summary

Solutions 0