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A348110
Number of positive integers <= n that have middle divisors.
2
1, 2, 2, 3, 3, 4, 4, 5, 6, 6, 6, 7, 7, 7, 8, 9, 9, 10, 10, 11, 11, 11, 11, 12, 13, 13, 13, 14, 14, 15, 15, 16, 16, 16, 17, 18, 18, 18, 18, 19, 19, 20, 20, 20, 21, 21, 21, 22, 23, 24, 24, 24, 24, 25, 25, 26, 26, 26, 26, 27, 27, 27, 28, 29, 29, 30, 30, 30, 30, 31
OFFSET
1,2
COMMENTS
a(n) is also the number of positive integers k <= n whose symmetric representation of sigma(k) has an odd number of parts.
EXAMPLE
For n = 9 there are six positive integers <= 9 that have middle divisors, they are [1, 2, 4, 6, 8, 9], so a(9) = 6.
On the other hand for n = 9 there are six positive integers k <= 9 whose symmetric representation of sigma(k) has an odd number of parts, they are [1, 2, 4, 6, 8, 9], so a(9) = 6.
MATHEMATICA
f[n_] := Boole[DivisorSum[n, 1 &, n/2 <= #^2 < 2*n &] > 0]; Accumulate@ Array[f, 100] (* Amiram Eldar, Oct 01 2021 *)
PROG
(PARI) f(n) = sumdiv(n, d, d2 = d^2; n / 2 < d2 && d2 <= n << 1); \\ A067742
a(n) = sum(k=1, n, f(k) > 0); \\ Michel Marcus, Oct 01 2021
KEYWORD
nonn
AUTHOR
Omar E. Pol, Sep 30 2021
STATUS
approved