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

1,1

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.

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.

Example

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

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 *)

Prog

(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<maxval) {
while(len<1000 & (k=x^2+2^y)<maxval) {
    sol[(len=len+1), ]=c(k, x, y)
    y=nextprime(y)
}
x=nextprime(x); y=2
}
sol=sol[1:len, ]; sort(unique(as.numeric(sol[, 1])))

Crossrefs

Cf. A000040, A001248, A034785.

Sequence in context: A331801 A143813 A290135 * A143845 A270888 A281014

Adjacent sequences: A225530 A225531 A225532 * A225534 A225535 A225536

Keyword

nonn

Author

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

Status

approved