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Number of unordered n-tuples {x_1, x_2, x_3, ..., x_n} such that Sum_{k=1..n} 1/x_k is an integer and x_k is an integer between 1 and n for 1 <= k <= n.
2

%I #39 Jul 15 2023 14:02:13

%S 1,1,2,3,6,9,25,39,84,158,381,610,2175,3489,7252,24744,54658,89031,

%T 273604,443746,1690517,5261990,9399018,15470605,58261863,102574465

%N Number of unordered n-tuples {x_1, x_2, x_3, ..., x_n} such that Sum_{k=1..n} 1/x_k is an integer and x_k is an integer between 1 and n for 1 <= k <= n.

%e 1/1 + 1/1 = 2 and 2 is an integer.

%e 1/1 + 1/2 = 3/2.

%e 1/2 + 1/2 = 1 and 1 is an integer.

%e So a(2) = 2.

%o (Ruby)

%o def A(n)

%o return 1 if n == 0

%o cnt = 0

%o (1..n).to_a.repeated_combination(n){|i|

%o cnt += 1 if (1..n).inject(0){|s, j| s + 1 / i[j - 1].to_r}.denominator == 1

%o }

%o cnt

%o end

%o def A349148(n)

%o (0..n).map{|i| A(i)}

%o end

%o p A349148(10)

%o (Python)

%o from math import lcm

%o from itertools import combinations_with_replacement

%o def A349148(n):

%o k = lcm(*range(2,n+1))

%o dlist = (k//d for d in range(1,n+1))

%o return sum(1 for d in combinations_with_replacement(dlist,n) if sum(d) % k == 0) # _Chai Wah Wu_, Nov 09 2021

%Y Cf. A349146.

%K nonn,more

%O 0,3

%A _Seiichi Manyama_, Nov 08 2021

%E a(16)-a(25) from _Alois P. Heinz_, Nov 08 2021