2021/01/18 by Ajit Singh, Singh, Ajit, Rupam Barman +1
Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2101.06900
openalex publication_date 2021/01/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Andrews' (k, i)-singular overpartition function Ck, i(n) counts the number of overpartitions of n in which no part is divisible by k and only parts ≡ ± i\pmodk may be overlined. In recent times, divisibility of C3ℓ, ℓ(n), C4ℓ, ℓ(n) and C6ℓ, ℓ(n) by 2 and 3 are studied for certain values of ℓ. In this article, we study divisibility of C3ℓ, ℓ(n), C4ℓ, ℓ(n) and C6ℓ, ℓ(n) by primes p≥ 5. For all positive integer ℓ and prime divisors p≥ 5 of ℓ, we prove that C3ℓ, ℓ(n), C4ℓ, ℓ(n) and C6ℓ, ℓ(n) are almost always divisible by arbitrary powers of p. For s∈ \3, 4, 6\, we next show that the set of those n for which Cs⋅ℓ, ℓ(n) \not≡ 0\pmodpik is infinite, where k is a positive integer satisfying pik-1≥ ℓ. We further improve a result of Gordon and Ono on divisibility of ℓ-regular partitions by powers of certain primes. We also improve a result of Ray and Chakraborty on divisibility of ℓ-regular overpartitions by powers of certain primes.