A097058 Numbers of the form p^2 + 2^p for p prime.

8, 17, 57, 177, 2169, 8361, 131361, 524649, 8389137, 536871753, 2147484609, 137438954841, 2199023257233, 8796093024057, 140737488357537, 9007199254743801, 576460752303426969, 2305843009213697673, 147573952589676417417, 2361183241434822611889

Offset: 1

Author

Parthasarathy Nambi, Sep 15 2004

Keywords

• nonn
• easy

Comments

For any n>=3, a(n) is divisible by 3. This follows from the following simple result, combined with the fact that A061725(n), n>=3, is divisible by 3: Let r>=5 be an odd integer such that r^2 + 2 is divisible by 3. Then r^2 + 2^i is divisible by 3 for any odd integer i>=3. In particular, r^2 + 2^r is divisible by 3. This contribution was inspired by Problem of the Month - Math Central, MP98 (problem for October 2010), which asks for all primes p such that 2^p + p^2 is also a prime. - Shai Covo (green355(AT)netvision.net.il), Nov 02 2010

Examples

			For example, the first two terms are 2^2 + 2^2 = 8, 3^2 + 2^3 = 17
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..200

Programs

Maple

a:= proc(n) local p; p:= ithprime(n); p^2+2^p end:
seq(a(n), n=1..25);  # Alois P. Heinz, May 15 2013

Mathematica

Table[ Prime[n]^2 + 2^Prime[n], {n, 16}] (* Robert G. Wilson v, Sep 15 2004 *)
#^2+2^#&/@Prime[Range[20]] (* Harvey P. Dale, Jul 12 2011 *)

PARI

forprime(p=2,61,print1(p^2+2^p,",")) \\ Klaus Brockhaus

Extensions

More terms from Klaus Brockhaus, Ray Chandler and Robert G. Wilson v, Sep 15 2004