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A341541
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a(n) is the number of steps to reach square 1 for a walk starting from square n along the shortest path on the square spiral board without stepping on any prime number. a(n) = -1 if such a path does not exist.
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4
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0, 1, 2, 1, 2, 1, 2, 1, 2, 3, 4, -1, 4, 3, 2, 3, 4, 19, 2, 17, 16, 15, 2, 3, 4, 5, 4, 5, 6, 11, 6, 5, 4, 3, 4, 5, 6, 19, 18, 19, 18, 17, 16, 15, 14, 13, 4, 5, 6, 7, 6, 5, 6, 9, 10, 11, 10, 9, 6, 5, 4, 5, 6, 7, 8, 9, 10, 17, 18, 19, 18, -1, 16, 15, 14, 13, 12
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OFFSET
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1,3
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COMMENTS
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Conjecture: There is no "island of two or more nonprimes" enclosed by primes on the square spiral board. If the conjecture is true, then numbers n such that a(n) = -1 are the terms in A341542.
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LINKS
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EXAMPLE
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The shortest paths for a(n) <= 20 are illustrated in the figure attached in Links section. If more than one path are available, the path through the smallest number is chosen as the shortest path.
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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
step_max = 20; L_last = [1]; L2 = L_last; L3 = [[1]]
for step in range(1, step_max + 1):
L1 = []
for j in range(0, len(L_last)):
m = L_last[j]; k = 0
while k <= 3 and isprime(m) == 0:
m_k = neib(m)[k]
if m_k not in L1 and m_k not in L2: L1.append(m_k)
k += 1
L2 += L1; L3.append(L1); L_last = L1
i = 1
while i:
if isprime(neib(i)[0])*isprime(neib(i)[1])*isprime(neib(i)[2])*isprime(neib(i)[3]) == 1: print(-1)
elif i not in L2: break
for j in range(0, len(L3)):
if i in L3[j]: print(j); break
i += 1
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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