A353152 Perfect powers that are divisible by 5.

25, 100, 125, 225, 400, 625, 900, 1000, 1225, 1600, 2025, 2500, 3025, 3125, 3375, 3600, 4225, 4900, 5625, 6400, 7225, 8000, 8100, 9025, 10000, 11025, 12100, 13225, 14400, 15625, 16900, 18225, 19600, 21025, 22500, 24025, 25600, 27000, 27225, 28900, 30625

Offset: 1

Author

Marco Ripà, Apr 26 2022

Keywords

• nonn
• easy

Comments

a(n) == 0 (mod 25).

a(n) == {0, 25, 75} (mod 100).

Examples

			225 is a term since 225 = 15^2 is a power of a multiple of 5.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000351, A011557.

Intersection of A001597 and A008587.

Cf. A075090 (even perfect powers).

Programs

Mathematica

Select[Range[32000], And[GCD @@ FactorInteger[#][[All, 2]] > 1, Divisible[#, 5]] &] (* Michael De Vlieger, Apr 30 2022 *)

PARI

isok(k) = ispower(k) && !(k%5); \\ Michel Marcus, Apr 27 2022

PARI

upto(n) = { my(res = List()); forstep(i = 5, sqrtint(n), 5, c = i; for(e = 2, logint(n, i), c*=i; listput(res, c) ) ); Set(res) } \\ David A. Corneth, Apr 27 2022

Formula

a(n) = (5*m)^k for some positive m and some k > 1. - David A. Corneth, Apr 28 2022

Sum_{n>=1} 1/a(n) = -Sum_{k>=2} mu(k)*zeta(k)/5^k = 0.0756933503... - Amiram Eldar, Jul 02 2022