A344481 Isolated single primes enclosed by four composites on square spiral board of odd numbers.

97, 157, 233, 257, 293, 307, 331, 337, 359, 367, 389, 397, 409, 439, 449, 479, 487, 499, 503, 563, 607, 613, 631, 653, 677, 683, 691, 709, 727, 743, 751, 761, 773, 853, 863, 887, 907, 911, 929, 937, 967, 971, 983, 1013, 1069, 1087, 1117, 1181, 1187, 1193, 1201

Offset

1,1

Links

Table of n, a(n) for n=1..51.

Ya-Ping Lu, Illustration of isolated single primes on square spiral board

Example

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

Prog

(Python)
from sympy import prime, isprime; from math import sqrt, ceil
def neib(m):
    n = int(ceil((sqrt(m)+1.0)/2.0)); L = [m, m, m, m]
    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
    L[0]+=1 if m<z2 else 8*n-5 if m<=z3 else -1 if m<=z4 else -8*n+9
    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
    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
    L[3]+=8*n-7 if m<=z2 else -1 if m<=z3 else -8*n+11 if m<z4 else 1
    return L
for i in range(2, 200):
    p = prime(i); L1 = [2*neib(int((p+1)/2))[j]-1 for j in range(4)]
    if sum(isprime(k) for k in L1) == 0: print(p)

Crossrefs

Cf. A341542.

Sequence in context: A000923 A142528 A139500 * A261376 A142094 A142212

Adjacent sequences: A344478 A344479 A344480 * A344482 A344483 A344484

Keyword

nonn

Author

Ya-Ping Lu, May 20 2021

Status

approved