A369793 a(n) is the number of occurrences of n in A063655.

0, 1, 1, 2, 1, 3, 2, 3, 3, 3, 2, 5, 3, 5, 5, 5, 4, 5, 5, 7, 6, 7, 5, 8, 6, 7, 7, 7, 6, 10, 7, 9, 8, 9, 9, 10, 8, 10, 9, 11, 8, 13, 9, 13, 11, 12, 11, 14, 13, 11, 12, 15, 10, 15, 13, 15, 13, 14, 12, 15, 12, 18, 16, 15, 15, 17, 13, 17

Offset: 1

Author

Adnan Baysal, Feb 07 2024

Keywords

• nonn

Comments

Construct a directed graph whose vertex set is the set of all positive integers, and a directed edge from k to n belongs to this graph iff A063655(k) = n. a(n) is the in-degree of the vertex n in this graph. As conjectured in A369110, it is also conjectured here that the only cycles in this graph are from 4 to itself and between 5 and 6.

Examples

			a(1) = 0 since 1 does not exist in A063655. This is also clear from the definition of A063655, because there is no integral rectangle with semiperimeter 1.
a(2) = 1 because there is only one integral rectangle of area 1 with a minimal semiperimeter 2, which is the 1 X 1 square. So 2 appears only once in A063655, which means a(2) = 1.
a(4) = 2, because only A063655(3) and A063655(4) have the value 4. For any n > 4, A063655(n) > 4, because A063655(n) > 2 * sqrt(n) > 2 * sqrt(4) = 4. Hence, 4 cannot appear in the rest of A063655.
		

Crossrefs

Cf. A063655, A369110, A056737.

Programs

Mathematica

a=1156;Table[Count[Table[2*Median[Divisors[m]], {m,a}] ,n],{n,Floor[2*Sqrt[a]]}] (* James C. McMahon, Mar 12 2024 *)

Python

from sympy import divisors
def A369793(n): return sum(1 for m in range(1,(n**2>>2)+1) if (d:=divisors(m))[((l:=len(d))-1)>>1]+d[l>>1]==n) # Chai Wah Wu, Mar 25 2024