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URL: https://oeis.org/A389764

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A389764
Even sub-perfect powers: odd perfect powers (not including 1) minus 1.
8
8, 24, 26, 48, 80, 120, 124, 168, 224, 242, 288, 342, 360, 440, 528, 624, 728, 840, 960, 1088, 1224, 1330, 1368, 1520, 1680, 1848, 2024, 2186, 2196, 2208, 2400, 2600, 2808, 3024, 3124, 3248, 3374, 3480, 3720, 3968, 4224, 4488, 4760, 4912, 5040, 5328, 5624, 5928
OFFSET
1,1
REFERENCES
Z. A. Melzak, Companion to Concrete Mathematics: Mathematical Techniques and Various Applications, John Wiley & Sons, New York, 1973, p. 88.
LINKS
Eugène Catalan, Note sur la sommation de quelques séries, Journal de Mathématiques Pures et Appliquées, Serie 1, Volume 7 (1842), pp. 1-12.
Junesang Choi, Multiple gamma functions and their applications, in: G. Milovanović and M. Rassias (eds.), Analytic Number Theory, Approximation Theory, and Special Functions: In Honor of Hari M. Srivastava, Springer New York, 2014, pp. 93-129. See eq. (65), p. 119.
Junesang Choi and Hari M. Srivastava, Series Involving the Zeta Functions and a Family of Generalized Goldbach-Euler Series, American Mathematical Monthly, Vol. 121, No. 3 (2014), pp. 229-236.
George Chrystal, Algebra: An Elementary Text-book for the higher classes of secondary schools and for colleges, Part II, London: A. & C. Black, 1906, p. 422, exercise 18.
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.
Niels Nielsen, Handbuch der theorie der gammafunktion, Teubner, Leipzig, 1906, section 23, pp. 58-59.
FORMULA
a(n) = A075109(n+1) - 1.
Sum_{n>=1} 1/a(n) = 1 - log(2) (A244009) (Euler, 1744).
MATHEMATICA
seq[lim_] := Union[Table[m^k - 1, {k, 2, Log2[lim + 1]}, {m, 3, Surd[lim + 1, k], 2}] // Flatten]; seq[6000]
PROG
(PARI) list(lim) = {my(s = List()); for(k = 2, logint(lim+1, 2), forstep(m = 3, sqrtnint(lim+1, k), 2, listput(s, m^k - 1))); Set(s); }
(Python)
from sympy import mobius, integer_nthroot
from oeis_sequences.OEISsequences import bisection
def A389764(n): return bisection(lambda x:int(n+x+sum(mobius(k)*((integer_nthroot(x+1, k)[0]+1>>1)-1) for k in range(2, (x+1).bit_length()))), n, n) # Chai Wah Wu, Oct 14 2025
CROSSREFS
Intersection of A005843 and A045542.
Complement of A389765 within A045542.
Sequence in context: A254448 A029607 A244030 * A375432 A374590 A382292
KEYWORD
nonn,easy
AUTHOR
Amiram Eldar, Oct 14 2025
STATUS
approved