A389764 Even sub-perfect powers: odd perfect powers (not including 1) minus 1.

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

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.

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.

Cf. A075109, A244009.

Sequence in context: A254448 A029607 A244030 * A375432 A374590 A382292

Adjacent sequences: A389761 A389762 A389763 * A389765 A389766 A389767

Keyword

nonn,easy

Author

Amiram Eldar, Oct 14 2025

Status

approved