A303297 List of middle divisors: for every positive integer that has middle divisors, add its middle divisors to the sequence.
1, 1, 2, 2, 3, 2, 3, 3, 4, 3, 5, 4, 3, 4, 5, 4, 6, 5, 4, 7, 5, 6, 4, 5, 7, 6, 5, 8, 6, 7, 5, 9, 6, 8, 7, 5, 6, 9, 7, 8, 6, 10, 7, 9, 8, 6, 11, 7, 10, 6, 8, 9, 7, 11, 8, 10, 9, 7, 12, 8, 11, 9, 10, 7, 13, 8, 12, 7, 9, 11, 10, 8, 13, 9, 12, 10, 11, 8, 14, 9, 13, 8, 10, 12, 15, 11, 9, 14, 8, 10, 13, 11, 12, 9, 15
Offset: 1
Examples
The middle divisor of 1 is 1, so a(1) = 1. The middle divisor of 2 is 1, so a(2) = 1. There are no middle divisors of 3. The middle divisor of 4 is 2, so a(3) = 2. There are no middle divisors of 5. The middle divisors of 6 are 2 and 3, so a(4) = 2 and a(5) = 3. There are no middle divisors of 7. The middle divisor of 8 is 2, so a(6) = 2. The middle divisor of 9 is 3, so a(7) = 3. There are no middle divisors of 10. There are no middle divisors of 11. The middle divisors of 12 are 3 and 4, so a(8) = 3 and a(9) = 4.
Links
- Michel Marcus, Table of n, a(n) for n = 1..6934
- Michael De Vlieger, Plot (n,d) at (x,y) for middle divisors d | n and n <= 2^16.
- Michael De Vlieger, Plot (n,d) at (x,y) for middle divisors d | n and n <= 345, labeling n, and showing composite prime powers in gold, squarefree composites in green, numbers neither squarefree nor composite in blue, and highlighting products of composite prime powers in large light blue.
- Michael De Vlieger, Plot (n,d) at (x,y) for middle divisors d | n and n <= 2^16, with same color function as above so as to show patterns according to prime power decomposition of n.
Crossrefs
Programs
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Mathematica
Table[Select[Divisors@ n, Sqrt[n/2] <= # < Sqrt[2 n] &] /. {} -> Nothing, {n, 135}] // Flatten (* Michael De Vlieger, Jun 14 2018 *) -
PARI
lista(nn) = {my(list = List()); for (n=1, nn, my(v = select(x->((x >= sqrt(n/2)) && (x < sqrt(n*2))), divisors(n))); for (i=1, #v, listput(list, v[i]));); Vec(list);} \\ Michel Marcus, Mar 26 2023
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