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A350936
a(n) is the smallest number m such that tau(m) = n*tau(m-1) = n*tau(m+1) or 0 if no such m exists, where tau(k) = A000005(k).
2
34, 6, 12, 30, 816, 60, 192, 270, 180, 240, 56320, 420, 233472, 2112, 1620, 1320, 2162688, 2340, 786432, 3120, 4800, 15360, 62914560, 3360, 172368, 724992, 6300, 29760, 24964497408, 12240, 35433480192, 7560, 599040, 15138816, 81648, 21600, 7215545057280
OFFSET
1,1
COMMENTS
Corresponding values of tau(a(n)): 4, 4, 6, 8, 20, 12, 14, 16, 18, 20, 44, 24, 52, 28, 30, 32, 68, 36, 38, 40, 42, 44, 92, 48, 100, 52, 54, 56, 116, 60, 124, 64, 132, 136, 70, 72, 296, ...
Triples of [tau(a(n) - 1), tau(a(n)), tau(a(n) + 1)] = [tau(a(n)) / n, tau(a(n)), tau(a(n)) / n]: [4, 4, 4], [2, 4, 2], [2, 6, 2], [2, 8, 2], [4, 20, 4], [2, 12, 2], [2, 14, 2], [2, 16, 2], [2, 18, 2], [2, 20, 2], [4, 44, 4], ...
EXAMPLE
a(3) = 12 because 12 is the smallest number m such that tau(m) = 3 * tau(m-1) = 3 * tau(m+1); tau(12) = 3 * tau(11) = 3 * tau(13) = 3 * 2 = 6.
PROG
(Magma) Ax:=func<n|exists(r){m: m in[2..10^6] | n * #Divisors(m - 1) eq n * #Divisors(m + 1) and n * #Divisors(m + 1) eq #Divisors(m)} select r else 0>; [Ax(n): n in [1..16]]
CROSSREFS
KEYWORD
nonn
AUTHOR
Jaroslav Krizek, Jan 25 2022
EXTENSIONS
a(23)-a(37) from Jon E. Schoenfield, Jan 25 2022
STATUS
approved