OFFSET
1,1
REFERENCES
William Dunham, Euler: The Master of Us All, The Mathematical Association of America, Washington D.C., 1999, p. 65.
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
Eugène Catalan, Note sur la sommation de quelques séries, Journal de Mathématiques Pures et Appliquées, Serie 1, Volume 7 (1842), pp. 1-12. See p. 6, eq. (16).
Leonhard Euler, Variae observationes circa series infinitas, Commentarii academiae scientiarum Petropolitanae, Vol. 9 (1744), pp. 160-188; reprinted in Opera Omnia, Series 1, Vol. 14, pp. 217-244.
Nicolao Fuss, Demonstratio Theorematum Quorundam Analyticorum, Nova Acta Academiae Scientiarum Imperialis Petropolitanae, Vol. 8 (1794), pp. 201-226. See pp. 223-226.
Joakim Munkhammar, 104.32 The Riemann zeta function as a sum of geometric series, The Mathematical Gazette, Vol. 104, No. 561 (2020), pp. 527-530.
Eric Weisstein's World of Mathematics, Perfect Power.
FORMULA
From Terry D. Grant, Oct 25 2020: (Start)
a(n) = A001597(n+1)^2 - 1.
Sum_{k>=1} 1/a(k) = 7/4 - Pi^2/6 = 7/4 - zeta(2).
Sum_{k>=1} 1/(a(k)+1) = Sum_{k>=2} mu(k)*(1-zeta(2*k)). (End)
EXAMPLE
a(2) = 63 because the perfect square 64 = 8^2 = 4^3.
MATHEMATICA
Take[ Select[ Range[ 2, 150 ], GCD@@(Last/@FactorInteger[ # ])>1& ]^2-1] (* corrected by Jon Maiga, Sep 28 2019 *)
PROG
(Python)
from sympy import mobius, integer_nthroot
def A062965(n):
def f(x): return int(n-1+x+sum(mobius(k)*(integer_nthroot(x, k)[0]-1) for k in range(2, x.bit_length())))
kmin, kmax = 1, 2
while f(kmax) >= kmax:
kmax <<= 1
while True:
kmid = kmax+kmin>>1
if f(kmid) < kmid:
kmax = kmid
else:
kmin = kmid
if kmax-kmin <= 1:
break
return kmax**2-1 # Chai Wah Wu, Aug 14 2024
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Jason Earls, Jul 16 2001
EXTENSIONS
More terms from Dean Hickerson, Jul 24 2001
STATUS
approved