A135172 - OEIS

%I #49 Sep 08 2022 08:45:32

%S 13,35,275,2315,179195,1602515,129271235,1162785755,94151567435,

%T 68630914235795,617675543767595,450284043329950835,

%U 36472998576194041955,328256976190630099835,26588814499694991643115,19383245676687219151537715,14130386092315195257068234555,127173474827954453552096993555

%N a(n) = 3^prime(n) + 2^prime(n).

%C 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.

%C 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

%H Vincenzo Librandi, <a href="/A135172/b135172.txt">Table of n, a(n) for n = 1..200</a>

%H H. Abbott, <a href="https://cms.math.ca/crux/backfile/Crux_v12n03_Mar.pdf">West German Mathematical Olympiad 1981 - First round, Problem 4</a>, Crux Mathematicorum, p. 42, Vol. 12, Mar. 86.

%H The IMO Compendium, <a href="https://imomath.com/othercomp/Ger/GerBW181.pdf">11-th German Federal Mathematical Competition 1980/81 - First round, Problem 4</a>.

%H <a href="/index/O#Olympiads">Index to sequences related to Olympiads</a>.

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

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

%p [3^ithprime(n)+2^ithprime(n)$n=1..20]; # _Muniru A Asiru_, Mar 05 2019

%t 3^# + 2^# &/@Prime[Range[20]] (* _Harvey P. Dale_, Dec 18 2010 *)

%o (Magma) [3^p+2^p: p in PrimesUpTo(100)]; // _Vincenzo Librandi_, Dec 14 2010

%o (Python)

%o from sympy import prime, primerange

%o def aupton(nn): return [3**p + 2**p for p in primerange(1, prime(nn)+1)]

%o print(aupton(18)) # _Michael S. Branicky_, Nov 21 2021

%Y Subsequence of A007916.

%Y Cf. A000040, A001597.

%K nonn,easy

%O 1,1

%A _Omar E. Pol_, Nov 25 2007

%E More terms from _Vincenzo Librandi_, Dec 14 2010