A353152 -id:A353152 - cp's OEIS frontend

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

Marco Ripà, May 02 2022

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).

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

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

Cf. A000244.
Intersection of A001597 and A008585.
Intersection of A001597 and A008591.
Other perfect powers: A075090 (even), A075109 (odd), A353152 (multiple of 5).

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

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

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

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

A351663 Perfect powers that are divisible by 7.

Original entry on oeis.org

49, 196, 343, 441, 784, 1225, 1764, 2401, 2744, 3136, 3969, 4900, 5929, 7056, 8281, 9261, 9604, 11025, 12544, 14161, 15876, 16807, 17689, 19600, 21609, 21952, 23716, 25921, 28224, 30625, 33124, 35721, 38416, 41209, 42875, 44100, 47089, 50176, 53361, 56644

Offset: 1

Marco Ripà, May 04 2022

Terms are multiples of 49, since no perfect power divisible by 7 can have a 7-adic valuation below 2.

			196 is a term since 196 = (2*7)^2 is a power of a multiple of 7.

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

Intersection of A001597 and A008589.

Other perfect powers: A075090, A075109, A353238, A353152.

q:= n-> igcd(seq(i[2], i=ifactors(n)[2]))>1:
select(q, [49*i$i=1..2000])[];  # Alois P. Heinz, May 05 2022

Select[49*Range[1200], GCD @@ FactorInteger[#][[All, 2]] > 1 &]

PARI
```
isok(k) = ispower(k) && !(k % 7)
```

a(n) has the form (7*m)^k for some m > 0 and k > 1.

Sum_{n>=1} 1/a(n) = -Sum_{k>=2} mu(k)*zeta(k)/7^k = 0.0371288923... - Amiram Eldar, Jul 02 2022

cp's OEIS Frontend

A353238 Perfect powers that are divisible by 3.

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