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

1,1

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).

Links

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

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

Example

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

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 &]

Prog

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

Crossrefs

Cf. A000244.

Intersection of A001597 and A008585.

Intersection of A001597 and A008591.

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

Sequence in context: A036303 A359030 A325299 * A116455 A103753 A238333

Adjacent sequences: A353235 A353236 A353237 * A353239 A353240 A353241

Keyword

nonn,easy

Author

Marco Ripà, May 02 2022

Status

approved