A353238 Perfect powers that are divisible by 3.

9, 27, 36, 81, 144, 216, 225, 243, 324, 441, 576, 729, 900, 1089, 1296, 1521, 1728, 1764, 2025, 2187, 2304, 2601, 2916, 3249, 3375, 3600, 3969, 4356, 4761, 5184, 5625, 5832, 6084, 6561, 7056, 7569, 7776, 8100, 8649, 9216, 9261, 9801, 10404, 11025, 11664, 12321

Offset: 1

Author

Marco Ripà, May 02 2022

Keywords

• nonn
• easy

Comments

Terms are multiples of 9, so that a(n) == 0 (mod 9) (since no perfect power divisible by 3 can have a 3-adic valuation below 2).

Examples

			36 is a term since 36 = (2*3)^2 is a power of a multiple of 3.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000244.

Intersection of A001597 and A008585.

Intersection of A001597 and A008591.

Other perfect powers: A075090 (even), A075109 (odd), A353152 (multiple of 5).

Programs

Maple

q:= n-> igcd(seq(i[2], i=ifactors(n)[2]))>1:
select(q, [9*i$i=1..2000])[];  # Alois P. Heinz, May 05 2022

Mathematica

Select[9*Range[1400], GCD @@ FactorInteger[#][[All, 2]] > 1 &]

PARI

isok(k) = ispower(k) && !(k % 3); \\ Michel Marcus, May 02 2022

Formula

a(n) has the form (3*m)^k for some positive integer m := m(n) and some k > 1.

Sum_{n>=1} 1/a(n) = -Sum_{k>=2} mu(k)*zeta(k)/3^k = 0.2306128559... - Amiram Eldar, Jul 02 2022