On a Recursively Defined Sequence Involving the Prime Counting Function

Journal of Integer Sequences, Vol. 24 (2021), Article 21.3.1

On a Recursively Defined Sequence Involving the Prime Counting Function

Altug Alkan
Graduate School of Science and Engineering
Piri Reis University
Istanbul
Turkey
Andrew R. Booker
School of Mathematics
University of Bristol
Woodland Road
Bristol, BS8 1UG
United Kingdom
Florian Luca
School of Mathematics
University of the Witwatersrand
Private Bag X3, Wits 2050
Johannesburg
South Africa
Research Group in Algebraic Structures and Applications
King Abdulaziz University
Jeddah
Saudi Arabia
Centro de Ciencias Matemáticas
UNAM
Morelia
Mexico

Abstract:

We prove some properties of sequence A335294 from the On-Line Encyclopedia of Integer Sequences, defined by a_n = π(n) - π (Σ_k=1ⁿ⁻¹ a_k), where π(x) is the number of primes ≤ x. In particular we show that the sequence (a_n) assumes every non-negative integral value infinitely often.

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(Concerned with sequences A335294 A335337.)

Received June 12 2020; revised version received January 27 2021. Published in Journal of Integer Sequences, January 31 2021.

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