A389765 Odd sub-perfect powers: even perfect powers minus 1.

3, 7, 15, 31, 35, 63, 99, 127, 143, 195, 215, 255, 323, 399, 483, 511, 575, 675, 783, 899, 999, 1023, 1155, 1295, 1443, 1599, 1727, 1763, 1935, 2047, 2115, 2303, 2499, 2703, 2743, 2915, 3135, 3363, 3599, 3843, 4095, 4355, 4623, 4899, 5183, 5475, 5775, 5831, 6083

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 pp. 118-119.

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, p. 59, eq. (11).

Formula

a(n) = A075090(n) - 1.

Sum_{n>=1} 1/a(n) = log(2) (A002162) (Euler, 1744).

Mathematica

seq[lim_] := Union[Table[m^k - 1, {k, 2, Log2[lim + 1]}, {m, 2, Surd[lim + 1, k], 2}] // Flatten]; seq[6000]

Prog

(PARI) list(lim) = {my(s = List());  for(k = 2, logint(lim+1, 2), forstep(m = 2, 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 A389765(n): return bisection(lambda x:int(n+x+sum(mobius(k)*(integer_nthroot(x+1, k)[0]>>1) for k in range(2, (x+1).bit_length()))), n, n) # Chai Wah Wu, Oct 14 2025

Crossrefs

Intersection of A005408 and A045542.

Complement of A389764 within A045542.

Cf. A075090, A002162.

Sequence in context: A090633 A320024 A098583 * A275531 A275532 A212315

Adjacent sequences: A389762 A389763 A389764 * A389766 A389767 A389768

Keyword

nonn,easy

Author

Amiram Eldar, Oct 14 2025

Status

approved