A062834
Denominator of sum of first n terms of the series 1/3 + 1/8 + 1/24 ... in which the denominators are one less than a perfect square that cannot otherwise be written as a power (cf. A062757, A037450).
Original entry on oeis.org
3, 24, 2, 70, 1680, 55440, 55440, 65520, 65520, 5040, 10080, 5040, 1627920, 1627920, 85680, 942480, 21677040, 492660, 107100, 64260, 621180, 1242360, 1328040, 10624320, 1328040, 14608440, 1217370, 277560360, 133506533160
Offset: 1
- W. Dunham, Euler: The Master of Us All, The Mathematical Association of America, Washington D.C., 1999, p. 66.
- L. Euler, "Variae observationes circa series infinitas," Opera Omnia, Ser. 1, Vol. 14, pp. 216-244.
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Table[ Denominator[ Plus@@(Take[ Select[ Range[ 2, 50 ], GCD@@(Last/@FactorInteger[ # ])==1& ]^2-1, k ]^-1) ], {k, 1, 29} ]
A062965
Positive numbers which are one less than a perfect square that is also another power.
Original entry on oeis.org
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
a(2) = 63 because the perfect square 64 = 8^2 = 4^3.
- 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.
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Take[ Select[ Range[ 2, 150 ], GCD@@(Last/@FactorInteger[ # ])>1& ]^2-1] (* corrected by Jon Maiga, Sep 28 2019 *)
-
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
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