A225533 Numbers expressible as p^2 + 2^q where p and q are primes.

8, 12, 13, 17, 29, 33, 36, 41, 53, 57, 81, 125, 129, 132, 137, 153, 173, 177, 201, 249, 293, 297, 321, 365, 369, 393, 417, 489, 533, 537, 561, 657, 845, 849, 873, 965, 969, 993, 1089, 1373, 1377, 1401, 1497, 1685, 1689, 1713, 1809, 1853, 1857, 1881, 1977, 2052

Offset: 1

Author

Kevin L. Schwartz and Christian N. K. Anderson, May 09 2013

Keywords

• nonn

Comments

a(n) ~ n^2. For n > 468, the formula .358*n^2.085 provides an estimate of a(n) accurate to within 11%.

a(10) = 57 is the first term that meets the criterion in two ways (5^2 + 2^5 and 7^2 + 2^3). In the first 10000 terms, there are 30 terms expressible in two ways, but none expressible in three ways.

Examples

			29 = 5^2 + 2^2, and both 5 and 2 are prime.
		

Links

• Christian N. K. Anderson, Table of n, a(n) for n = 1..10000
• Christian N. K. Anderson, Ulam Spiral of a(n) for n = 1..10000.
• Christian N. K. Anderson, Table of n, a(n), and all values of (p,q) producing a(n) for n = 1..10000.

Crossrefs

Cf. A000040, A001248, A034785.

Programs

Mathematica

nn = 15; ps = Prime[Range[nn]]; p2 = Prime[Range[PrimePi[2*Log[2, ps[[-1]]]]]]; t = Table[p^2 + 2^q, {p, ps}, {q, p2}]; Union[Select[Flatten[t], # < ps[[-1]]^2 &]] (* T. D. Noe, May 15 2013 *)

R

library(gmp); x=y=as.bigz(2); maxval=10000; sol=as.bigz(matrix(0,nc=3,nr=1000)); len=0
while(len<1000 & x^2+2^y