Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.

Updated Oct 8

Problem 69

claims/: The 2 claim pages of Problem 69, one per claimant's result; the problem's standing derives from them.

Statement. Is

\sum_{n\geq 2}\frac{\omega(n)}{2^n}

irrational? (Here $\omega(n)$ counts the number of distinct prime divisors of $n$ .)

Status. Proved. Tao and Teräväinen [TaTe25] prove unconditionally that the series, which equals $\sum_p 1/(2^p-1)$ and is thus the case of Problem 257 with the primes as the infinite set, is irrational; Pratt [Pr24] had proved it under a uniform quantitative prime tuples conjecture. The accepted claim is Tao and Teräväinen; the conditional result is Pratt.

Source. erdosproblems.com/69, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #69, https://www.erdosproblems.com/69.

References.

[Er48] Erdős, P., On arithmetical properties of Lambert series. J. Indian Math. Soc. (N.S.) (1948), 63-66.
[Pr24] Pratt, K., The irrationality of a prime factor series under a prime tuples conjecture. arXiv:2409.15185 (2024).
[TaTe25] T. Tao and J. Teräväinen, Quantitative correlations and some problems on prime factors of consecutive integers. arXiv:2512.01739 (2025).

Formalization. Statement in formal-conjectures.

Current assessment

The site's formulation of 2026-10-07 asks whether $\sum_{n\geq2}\omega(n)/2^n$ is irrational. Answered yes: [[problems/irrationality/E0069/claims/2025_12_01_tao_teravainen|Tao and Teräväinen's claim page]] records Theorem 1.3 of [TaTe25], which proves the series, equal to $\sum_p 1/(2^p-1)$ and so the case of Problem 257 with the primes as the infinite set, irrational, a result the site's curator credits. Pratt's earlier theorem, that $\sum_n\omega(n)/t^n$ is irrational for every integer $t\geq2$ under a uniform quantitative prime tuples conjecture, is on Pratt's claim page; it is conditional and decides nothing for the problem's standing. Erdős proved in [Er48] that $\sum_n\tau(n)/2^n$ is irrational and wrote that the analogous series for the number of prime factors seemed to present difficulties. Status search of 2026-10-07: the site's page and remarks, its forum thread, which has no comments, the formal-conjectures file, and Crossref for a journal version of [TaTe25]; no refereed publication of the unconditional proof was found. The corpus holds no compiled or reviewed proof.

Linked library material

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Linked from (18)

Pratt's conditional irrationality under a prime tuples conjecture Tao and Teräväinen prove the prime-factor series irrational problems/irrationality/E0069/claims Problem 249 Problem 250 Irrationality and Diophantine Approximation arithmetic_functions/tao_2025_quantitative_correlations_problems_prime_factors_consecutive Irrationality and Diophantine Approximation irrationality/erdos_1948_arithmetical_properties_lambert_series irrationality/erdos_1957_irrationality_certain_series Remark on p. 212: the totient, divisor-sum and prime-factor series irrationality/erdos_1969_irrationality_certain_series irrationality/erdos_1988_irrationality_certain_series_problems_results irrationality/pratt_2024_irrationality_prime_factor_series_under_prime Conjecture 1.2: a uniform quantitative prime K-tuples conjecture Proposition 2.1: under Conjecture 1.2, some n_0 ≤ x makes n_0Q/k + 1 prime for k ≤ K with controlled ω beyond Theorem 1.3: under Conjecture 1.2, the sum of ω(n)/t^n is irrational for every integer t ≥ 2 number_theory/erdos_1980_old_new_problems_results_combinatorial_number_theory

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