Primes p such that p + 2^k is composite for all k >= 0Maximal number of squares of side 1 in an ellipse of semiaxes n,2n.
2131, 7013, 8543, 10711, 140333, 17, 43, 83, 137, 203, 279, 369, 471, 587, 715, 857, 1011, 1175, 1351, 1541, 1743, 1961, 2191, 2429, 2683, 2949, 3227, 3523, 3829, 4137, 4469, 4809, 5167, 5539, 5913, 6295, 6701, 7127, 7555, 7999, 8449, 8909, 9395, 9889, 10395, 10915
The example given suggests that these values are merely conjectures. - njas, Jun 30 2005
For small n there exists a k such that prime(n) + 2^k is prime however as n becomes large there are primes prime(n) that "defy" finding a k such that prime(n) + 2^k is prime. Defiant primes base 4 are much more frequent. Conjecture: The number of defiant primes base 2 is infinite.
For p = 2131 p + 2^k is not prime for all k < 20000
a(1)=3 since you cannot pack more than 3 unit-side squares in an ellipse of semiaxes 1,2
f[n_] := 2 Sum[IntegerPart[2 Sqrt[4 n^2 - (h - 1/2)^2]],
{h, 2, 2 n}] + IntegerPart[2 Sqrt[4 n^2 - 1/4]]; Array[f, 42]
(PARI) pplus2ton(n, m, b) = \defiant primes base b { local(k, s, p, y, flag); s=0; forprime(p=2, n, flag=1; for(k=0, m, y=p+b^k; if(ispseudoprime(y), \ print1(k, ", "); s++; flag=0; break) ); if(flag, print(p)); \search for defiant primes. ); print(); print(s); }
Similar to A125228.
nonn,new
easy,nonn
Cino Hilliard Pasquale CUTOLO (hillcino368p.cutolo(AT)gmailinwind.comit), Jun 27 200514 2007