TY - JOUR

T1 - On a Diophantine equation of Ayad and Kihel

AU - El Bachraoui, Mohamed

AU - Luca, Florian

N1 - Funding Information:
Acknowledgements. We thank the referees for comments which improved the quality of this paper. The second author worked on this paper during a visit to the Department of Mathematical Sciences of the United Arab Emirates University in Al-Ain in May 2011. He thanks the people of this institution for their hospitality. This paper was written in Spring of 2011 while he was in sabbatical from the Mathematical Institute UNAM from January 1 to June 30, 2011, and supported by a PASPA fellowship from DGAPA.

PY - 2012/6

Y1 - 2012/6

N2 - Let f(n) denote the number of relatively prime sets in {1,..., n}. This is sequence A085945 in Sloane's On-Line Encyclopedia of Integer Sequences. Motivated by a paper of Ayad and Kihel [1], we show that there are at most finitely many positive integers n such that f(n) is a perfect power of exponent > 1 of some other integer. We also show that the sequence {f(n)} n≥1 is not holonomic; that is, it satisfies no recurrence relation of finite order with polynomial coefficients.

AB - Let f(n) denote the number of relatively prime sets in {1,..., n}. This is sequence A085945 in Sloane's On-Line Encyclopedia of Integer Sequences. Motivated by a paper of Ayad and Kihel [1], we show that there are at most finitely many positive integers n such that f(n) is a perfect power of exponent > 1 of some other integer. We also show that the sequence {f(n)} n≥1 is not holonomic; that is, it satisfies no recurrence relation of finite order with polynomial coefficients.

KW - Prime subsets

KW - holonomic sequences

KW - perfect powers

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U2 - 10.2989/16073606.2012.697265

DO - 10.2989/16073606.2012.697265

M3 - Article

AN - SCOPUS:84863494357

VL - 35

SP - 235

EP - 243

JO - Quaestiones Mathematicae

JF - Quaestiones Mathematicae

SN - 1607-3606

IS - 2

ER -