OFFSET
1,2
COMMENTS
a(n) exist for all n>=1.
The solution k to A018804(k)/k = n is unique if and only if n is a power of 2: a(2^m) = 2^(2^(m+1)-2) = A051191(m+1).
The questions of existence and uniqueness are a part of a problem that was proposed during the Forty-Fifth International Mathematical Olympiad in Athens, Greece, July 7-19, 2004.
REFERENCES
Dušan Djukić, Vladimir Janković, Ivan Matić, and Nikola Petrović, The IMO Compendium, A Collection of Problems Suggested for the International Mathematical Olympiads: 1959-2004, Springer, New York, 2006. See Problem 25, pp. 331, 726-727.
LINKS
Dušan Djukić, Vladimir Janković, Ivan Matić, and Nikola Petrović, IMO Shortlist 2004, 2005, Problem 25, pp. 5-6, 18-19.
R. E. Woodrow, Problem N2, The Olympiad Corner, Crux Mathematicorum, Vol. 34, No. 7 (2008), pp. 419-421.
EXAMPLE
a(2) = 4 since A018804(4)/4 = 8/4 = 2 and 4 is the least number with this property.
MATHEMATICA
f[p_, e_] := (e*(p - 1)/p + 1); r[n_] := Times @@ (f @@@ FactorInteger[n]); a[n_] := Module[{k = 1}, While[r[k] != n, k++]; k]; Array[a, 15]
CROSSREFS
KEYWORD
nonn
AUTHOR
Amiram Eldar, Apr 09 2022
STATUS
approved