A135172 a(n) = 3^prime(n) + 2^prime(n).

13, 35, 275, 2315, 179195, 1602515, 129271235, 1162785755, 94151567435, 68630914235795, 617675543767595, 450284043329950835, 36472998576194041955, 328256976190630099835, 26588814499694991643115, 19383245676687219151537715, 14130386092315195257068234555, 127173474827954453552096993555

Offset

1,1

Comments

Only a(1) = 2^2 + 3^2 = 13 is prime. Since all other primes p are odd, hence of the form 2k+1, we have 2^(2k+1) + 3^(2k+1) is always divisible by 5, and is at best semiprime, such as 3^83 + 2^83 = 3990838394187349600940803592605746684635 = 5 * 798167678837469920188160718521149336927.

a(n) is never a perfect power (A001597), this question was asked during West Germany Olympiad in 1981 (see links). - Bernard Schott, Mar 05 2019

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..200

H. Abbott, West German Mathematical Olympiad 1981 - First round, Problem 4, Crux Mathematicorum, p. 42, Vol. 12, Mar. 86.

The IMO Compendium, 11-th German Federal Mathematical Competition 1980/81 - First round, Problem 4.

Index to sequences related to Olympiads.

Formula

a(n) = 3^A000040(n) + 2^A000040(n).

Example

a(4)=2315 because the 4th prime number is 7, 3^7=2187, 2^7=128 and 2187+128=2315.

Maple

[3^ithprime(n)+2^ithprime(n)$n=1..20]; # Muniru A Asiru, Mar 05 2019

Mathematica

3^# + 2^# &/@Prime[Range[20]] (* Harvey P. Dale, Dec 18 2010 *)

Prog

(Magma) [3^p+2^p: p in PrimesUpTo(100)]; // Vincenzo Librandi, Dec 14 2010
(Python)
from sympy import prime, primerange
def aupton(nn): return [3**p + 2**p for p in primerange(1, prime(nn)+1)]
print(aupton(18)) # Michael S. Branicky, Nov 21 2021

Crossrefs

Subsequence of A007916.

Cf. A000040, A001597.

Sequence in context: A242578 A192145 A221592 * A183309 A272108 A034119

Adjacent sequences: A135169 A135170 A135171 * A135173 A135174 A135175

Keyword

nonn,easy

Author

Omar E. Pol, Nov 25 2007

Extensions

More terms from Vincenzo Librandi, Dec 14 2010

Status

approved