OFFSET
0,2
COMMENTS
a(n) is the number of terms in A341541 whose value equals n.
If stepping on prime squares is permitted, a(n) = 4*n. Conjecture: lim_{n->oo} a(n)/n = 4.
PROG
(Python)
from sympy import prime, isprime
from math import sqrt, ceil
def neib(m):
if m == 1: L = [4, 6, 8, 2]
else:
n = int(ceil((sqrt(m) + 1.0)/2.0))
z1 = 4*n*n - 12*n + 10; z2 = 4*n*n - 10*n + 7; z3 = 4*n*n - 8*n + 5; z4 = 4*n*n - 6*n + 3; z5 = 4*n*n - 4*n + 1
if m == z1: L = [m + 1, m - 1, m + 8*n - 9, m + 8*n - 7]
elif m > z1 and m < z2: L = [m + 1, m - 8*n + 15, m - 1, m + 8*n - 7]
elif m == z2: L = [m + 8*n - 5, m + 1, m - 1, m + 8*n - 7]
elif m > z2 and m < z3: L = [m + 8*n - 5, m + 1, m - 8*n + 13, m - 1]
elif m == z3: L = [m + 8*n - 5, m + 8*n - 3, m + 1, m - 1]
elif m >z3 and m < z4: L = [m - 1, m + 8*n - 3, m + 1, m - 8*n + 11]
elif m == z4: L = [m - 1, m + 8*n - 3, m + 8*n - 1, m + 1]
elif m > z4 and m < z5: L = [m - 8*n + 9, m - 1, m + 8*n - 1, m + 1]
elif m == z5: L = [m - 8*n + 9, m - 1, m + 8*n - 1, m + 1]
return L
print(1)
L_1 = [1]; L_in = [1]; step_max = 100
for step in range(1, step_max + 1):
L = []
for j in range(0, len(L_1)):
m = L_1[j]
if isprime(m) == 0:
for k in range(4):
m_k = neib(m)[k]
if m_k not in L_in: L.append(m_k); L_in.append(m_k)
print(len(L))
L_1 = L
CROSSREFS
KEYWORD
nonn
AUTHOR
Ya-Ping Lu, Feb 17 2021
STATUS
approved