A225592 a(n) = size of the network of triangular(n). Distinct triangular numbers T and R are directly connected if R = T*k, and k>0 is a triangular number less than T. Numbers belong to the same network if there is a chain of direct connections between them.

1, 1, 1, 1, 2, 7, 1, 1, 2, 8, 2, 3, 7, 7, 1, 1, 1, 5, 4, 7, 2, 3, 4, 1, 1, 1, 1, 2, 2, 1, 1, 1, 3, 8, 7, 2, 4, 6, 3, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 8, 1, 1, 1, 2, 1, 4, 1, 2, 2, 1, 1, 1, 3, 2, 3, 1, 1, 1, 1, 4, 3, 2, 5, 2, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 7, 2

Offset: 1

Author

Alex Ratushnyak, May 11 2013

Keywords

• nonn

Examples

			Triangular(6) = 21 belongs to the network with the following members: 21, 105, 210, 630, 19110, 25200, 145530. So a(6) = 7.
Triangular(133) = 8911 belongs to the network with the following members: 8911, 810901, 28158760, 3104129028, 328779810450, 543633020560, 18474540054600, 45742477468287600, 1553903798634573750. So a(133) = 9.
		

Crossrefs

Cf. A000217, A188630.

Programs

Python

def isTriangular(a):
    sr = 1 << (a.bit_length() >> 1)
    a += a
    while a < sr*(sr+1):  sr>>=1
    b = sr>>1
    while b:
      s = sr+b
      if a >= s*(s+1):  sr = s
      b>>=1
    return (a==sr*(sr+1))
network = [0]*1000
readPos = writePos = 0
def addConnection(R):
  global writePos
  if isTriangular(R):
    for j in range(readPos):
      if network[j]==R:  return
    network[writePos] = R
    writePos += 1
for i in range(1, 1000000):
    readPos, writePos = 0, 1
    network[0] = i*(i+1)//2
    while readPos < writePos:
      T = network[readPos]
      readPos += 1
      n = k = 3
      while k < T:
        addConnection(T*k)
        if T % k == 0 and T//k > k:  addConnection(T//k)
        k += n
        n += 1
    print(readPos, end=', ')