A390156 Numbers of the form (3*m + 1)^k - 1, where m >= 1 and k >= 2.

15, 48, 63, 99, 168, 255, 342, 360, 483, 624, 783, 960, 999, 1023, 1155, 1368, 1599, 1848, 2115, 2196, 2400, 2703, 3024, 3363, 3720, 4095, 4488, 4899, 5328, 5775, 6240, 6723, 6858, 7224, 7743, 8280, 8835, 9408, 9999, 10608, 10647, 11235, 11880, 12543, 13224, 13923

Offset

1,1

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

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 section 5.1, p. 118.

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.

Formula

a(n) == 0 (mod 3).

Sum_{n>=1} 1/a(n) = 1 - log(3)/2 - Pi/(6*sqrt(3)) = 1 - A156057 - A381671 = 0.1483939... .

Example

63 is a term since (3*1 + 1)^3 - 1 = 63.

Mathematica

seq[lim_] := Union[Table[m^k - 1, {k, 2, Log2[lim + 1]}, {m, 4, Surd[lim + 1, k], 3}] // Flatten]; seq[14000]

Prog

(PARI) list(lim) = {my(s = List());  for(k = 2, logint(lim+1, 2), forstep(m = 4, sqrtnint(lim+1, k), 3, listput(s, m^k - 1))); Set(s); }
(Python)
from sympy import mobius, integer_nthroot
from oeis_sequences.OEISsequences import bisection
def A390156(n):
    def f(x): return n+x+sum(mobius(k)*((integer_nthroot(x+1, k)[0]-1)//3) for k in range(2, (x+1).bit_length()+1>>1))
    return bisection(f, n, n) # Chai Wah Wu, Oct 31 2025

Crossrefs

Complement of the disjoint union of A390155 and A390157 within A045542.

Cf. A001597, A156057, A353238, A381671, A389764, A389765.

Sequence in context: A000813 A156205 A065906 * A370912 A377521 A154060

Adjacent sequences: A390153 A390154 A390155 * A390157 A390158 A390159

Keyword

nonn,easy

Author

Amiram Eldar, Oct 27 2025

Status

approved