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

1,1

Comments

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

Links

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

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

Example

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

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 &]

Prog

(PARI) isok(k) = ispower(k) && !(k % 7)
(Python)
from sympy import mobius, integer_nthroot
from oeis_sequences.OEISsequences import bisection
def A351663(n):
    def f(x): return int(n+x+sum(mobius(k)*(integer_nthroot(x, k)[0]//7) for k in range(2, x.bit_length())))
    return bisection(f, n, n) # Chai Wah Wu, Sep 18 2025

Crossrefs

Intersection of A001597 and A008589.

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

Sequence in context: A251214 A158638 A198386 * A016982 A157919 A297895

Adjacent sequences: A351660 A351661 A351662 * A351664 A351665 A351666

Keyword

nonn,easy

Author

Marco Ripà, May 04 2022

Status

approved