A369110 -id:A369110 - cp's OEIS frontend

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

Adnan Baysal, Feb 07 2024

nonn

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.

			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.

Cf. A063655, A369110, A056737.

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

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

A371597 a(n) is the sum of k where A063655(k) = n.

Original entry on oeis.org

0, 1, 2, 7, 6, 22, 22, 38, 52, 70, 58, 141, 104, 188, 230, 281, 260, 320, 374, 531, 526, 717, 566, 927, 756, 1017, 1114, 1203, 1148, 1799, 1402, 1741, 1718, 2170, 2314, 2765, 2400, 2912, 2800, 3769, 2856, 4577, 3352, 4923, 4410, 5054, 5036, 6346, 6246, 5537

Offset: 1

Adnan Baysal, Mar 28 2024

nonn

Construct the same directed graph as in A369793. a(n) is the sum of vertices directed to the vertex n in this graph.

			a(1) = 0 since 1 does not exist in A063655.
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 at index 1, which means a(2) = 1.
a(4) = 7, 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.

Cf. A056737, A063655, A369110, A369793.

from sympy import divisors
def A371597(n): return sum(m for m in range(1, (n**2>>2)+1) if (d:=divisors(m))[((l:=len(d))-1)>>1]+d[l>>1]==n)

cp's OEIS Frontend

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

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