Claims

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1952_06_01_erdos: Erdős's Monthly Problem 4493 of 1952, solved in print by J. B. Kelly in 1953, states that the sum of sigma(n)/n! is irrational, the case k = 1.

1953_01_01_erdos_kac: Erdős and Kac's Monthly Problem 4518 of 1953, solved in print by R. Breusch in 1954, states that the sum of sigma_2(n)/n! is irrational, the case k = 2; every later paper on the series cites it as settled there.

1971_03_01_erdos_straus: Erdős and Straus's general theorems on series over products a_1 through a_n (1971, Theorem 2.26; 1974, Theorem 3.7) give, with a_n = n, that the sum of sigma(n)/n! is irrational, the case k = 1.

2006_12_01_schlage_puchta: Schlage-Puchta's 2006 paper proves by a sieve count and exponential-sum estimates that the sum of sigma_3(n)/n! is irrational, the case k = 3, independently of Friedlander, Luca and Stoiciu; refereed in Ramanujan J.

2006_12_01_schlage_puchta_conditional: Schlage-Puchta's 2006 paper proves that Schinzel's Hypothesis H implies that the sum of sigma_k(n)/n! is irrational for every k; the hypothesis is unproven, so the page derives nothing for the problem's standing.

2007_07_03_friedlander_luca_stoiciu: Friedlander, Luca and Stoiciu's 2007 paper proves by a Chen-type sieve theorem that the sum of sigma_3(n)/n! is irrational, the case k = 3, independently of Schlage-Puchta; refereed in Integers.

2007_07_03_friedlander_luca_stoiciu_conditional: Friedlander, Luca and Stoiciu's 2007 paper proves that the prime k-tuples conjecture in Dickson's form implies that the sum of sigma_k(n)/n! is irrational for every k; the hypothesis is unproven.

2022_09_22_pratt: Pratt's paper, posted in 2022 and published in Acta Arithmetica in 2023, proves by sieve methods and exponential sums that the sum of sigma_4(n)/n! is irrational, the case k = 4.

2026_09_08_tokengrinder: A kernel-checked Lean 4 development, published anonymously in September 2026 under the name Tokengrinder, proves that the sum of sigma_k(n)/n! is irrational for every natural k; rebuilt and statement-audited here.

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