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A062965
Positive numbers which are one less than a perfect square that is also another power.
3
15, 63, 80, 255, 624, 728, 1023, 1295, 2400, 4095, 6560, 9999, 14640, 15624, 16383, 20735, 28560, 38415, 46655, 50624, 59048, 65535, 83520, 104975, 117648, 130320, 159999, 194480, 234255, 262143, 279840, 331775, 390624, 456975, 531440, 614655
OFFSET
1,1
REFERENCES
William Dunham, Euler: The Master of Us All, The Mathematical Association of America, Washington D.C., 1999, p. 65.
Leonhard Euler, "Variae observationes circa series infinitas," Opera Omnia, Ser. 1, Vol. 14, pp. 216-244.
Nicolao Fvss, "Demonstratio Theorematvm Qvorvndam Analyticorvm," Nova Acta Academiae Scientiarum Imperialis Petropolitanae, 8 (1794) 223-226.
LINKS
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
AUTHOR
Jason Earls, Jul 16 2001
EXTENSIONS
More terms from Dean Hickerson, Jul 24 2001
STATUS
approved