A351663 Perfect powers that are divisible by 7.

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

Author

Marco Ripà, May 04 2022

Keywords

• nonn
• easy

Comments

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

Examples

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

Links

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

Crossrefs

Intersection of A001597 and A008589.

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

Programs

Maple

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

Mathematica

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

PARI

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

Formula

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