A344481 - OEIS

%I #6 Jun 24 2021 21:38:04

%S 97,157,233,257,293,307,331,337,359,367,389,397,409,439,449,479,487,

%T 499,503,563,607,613,631,653,677,683,691,709,727,743,751,761,773,853,

%U 863,887,907,911,929,937,967,971,983,1013,1069,1087,1117,1181,1187,1193,1201

%N Isolated single primes enclosed by four composites on square spiral board of odd numbers.

%H Ya-Ping Lu, <a href="/A344481/a344481.pdf">Illustration of isolated single primes on square spiral board</a>

%e 3 is not a term because two of the four neighbors (1, 5, 17 and 21) are primes.

%e 97 is a term because 97 is a prime and all four neighbors (51, 95, 99 and 159) are composites (see the illustration in Links).

%o (Python)

%o from sympy import prime, isprime; from math import sqrt, ceil

%o def neib(m):

%o     n = int(ceil((sqrt(m)+1.0)/2.0)); L = [m,m,m,m]

%o     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     L[0]+=1 if m<z2 else 8*n-5 if m<=z3 else -1 if m<=z4 else -8*n+9

%o     L[1]+=-1 if m==z1 else -8*n+15 if m<z2 else 1 if m<z3 else 8*n-3 if m<=z4 else -1

%o     L[2]+=8*n-9 if m==z1 else -1 if m<=z2 else -8*n+13 if m<z3 else i if m<z4 else 8*n-1

%o     L[3]+=8*n-7 if m<=z2 else -1 if m<=z3 else -8*n+11 if m<z4 else 1

%o     return L

%o for i in range(2, 200):

%o     p = prime(i); L1 = [2*neib(int((p+1)/2))[j]-1 for j in range(4)]

%o     if sum(isprime(k) for k in L1) == 0: print(p)

%Y Cf. A341542.

%K nonn

%O 1,1

%A _Ya-Ping Lu_, May 20 2021