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A341672 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.

%I A341672 #8 Feb 18 2021 00:48:02
%S A341672 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,
%T A341672 91,94,100,106,110,111,110,119,126,131,137,139,138,143,153,154,144,
%U A341672 152,144,152,156,170,195,193,193,192,198,203,215,215,209,216,222,225
%N A341672 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.
%C A341672 a(n) is the number of terms in A341541 whose value equals n.
%C A341672 If stepping on prime squares is permitted, a(n) = 4*n. Conjecture: lim_{n->oo} a(n)/n = 4.
%o A341672 (Python)
%o A341672 from sympy import prime, isprime
%o A341672 from math import sqrt, ceil
%o A341672 def neib(m):
%o A341672     if m == 1: L = [4, 6, 8, 2]
%o A341672     else:
%o A341672         n = int(ceil((sqrt(m) + 1.0)/2.0))
%o A341672         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
%o A341672         if m == z1:             L = [m + 1, m - 1, m + 8*n - 9, m + 8*n - 7]
%o A341672         elif m > z1 and m < z2: L = [m + 1, m - 8*n + 15, m - 1, m + 8*n - 7]
%o A341672         elif m == z2:           L = [m + 8*n - 5, m + 1, m - 1, m + 8*n - 7]
%o A341672         elif m > z2 and m < z3: L = [m + 8*n - 5, m + 1, m - 8*n + 13, m - 1]
%o A341672         elif m == z3:           L = [m + 8*n - 5, m + 8*n - 3, m + 1, m - 1]
%o A341672         elif m >z3 and m < z4:  L = [m - 1, m + 8*n - 3, m + 1, m - 8*n + 11]
%o A341672         elif m == z4:           L = [m - 1, m + 8*n - 3, m + 8*n - 1, m + 1]
%o A341672         elif m > z4 and m < z5: L = [m - 8*n + 9, m - 1, m + 8*n - 1, m + 1]
%o A341672         elif m == z5:           L = [m - 8*n + 9, m - 1, m + 8*n - 1, m + 1]
%o A341672     return L
%o A341672 print(1)
%o A341672 L_1 = [1]; L_in = [1]; step_max = 100
%o A341672 for step in range(1, step_max + 1):
%o A341672     L = []
%o A341672     for j in range(0, len(L_1)):
%o A341672         m = L_1[j]
%o A341672         if isprime(m) == 0:
%o A341672             for k in range(4):
%o A341672                 m_k = neib(m)[k]
%o A341672                 if m_k not in L_in: L.append(m_k); L_in.append(m_k)
%o A341672     print(len(L))
%o A341672     L_1 = L
%Y A341672 Cf. A341541, A341542.
%K A341672 nonn
%O A341672 0,2
%A A341672 _Ya-Ping Lu_, Feb 17 2021