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A225533 Numbers expressible as p^2 + 2^q where p and q are primes. 1
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 (list; graph; refs; listen; history; text; internal format)
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
Sequence in context: A331801 A143813 A290135 * A143845 A270888 A281014
KEYWORD
nonn
AUTHOR
STATUS
approved

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Last modified March 28 10:55 EDT 2024. Contains 371241 sequences. (Running on oeis4.)