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A341672
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a(n) is the number of numbers on the square spiral board such that it takes n steps for them to reach square 1 along the shortest path without stepping on any prime number.
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3
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1, 4, 7, 5, 9, 8, 12, 10, 14, 23, 29, 32, 35, 38, 46, 47, 52, 59, 65, 64, 67, 76, 78, 84, 90, 91, 94, 100, 106, 110, 111, 110, 119, 126, 131, 137, 139, 138, 143, 153, 154, 144, 152, 144, 152, 156, 170, 195, 193, 193, 192, 198, 203, 215, 215, 209, 216, 222, 225
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OFFSET
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0,2
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COMMENTS
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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.
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LINKS
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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