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
SN - 1607-3606
VL - 35
SP - 235
EP - 243
JO - Quaestiones Mathematicae
JF - Quaestiones Mathematicae
IS - 2
ER -