A390157 Numbers of the form (3*m + 2)^k - 1, where m >= 0 and k >= 2, that are not of the form (3*m' + 1)^k' - 1 for any m' >= 1 and k' >= 2.

3, 7, 24, 31, 120, 124, 127, 195, 288, 399, 511, 528, 675, 840, 1224, 1330, 1443, 1680, 1935, 2047, 2208, 2499, 2743, 2808, 3124, 3135, 3480, 3843, 4224, 4623, 4912, 5040, 5475, 5928, 6399, 6888, 7395, 7920, 7999, 8191, 8463, 9024, 9603, 10200, 10815, 11448, 12099

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

Sum_{n>=1} 1/a(n) = Pi/(3*sqrt(3)) = A073010.

Example

63 = (3*0 + 2)^6 - 1 is not a term since 63 also equals (3*1 + 1)^3 - 1.
511 is a term since 511 = (3*0 + 2)^9 - 1 = (3*2 + 2)^3 - 1.

Mathematica

seq[lim_] := Complement[Union[Table[m^k - 1, {k, 2, Log2[lim + 1]}, {m, 2, Surd[lim + 1, k], 3}] // Flatten], Union[Table[m^k - 1, {k, 2, Log2[lim + 1]}, {m, 4, Surd[lim + 1, k], 3}] // Flatten]]; seq[13000]

Prog

(PARI) list1(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); }
list2(lim) = {my(s = List());  for(k = 2, logint(lim+1, 2), forstep(m = 2, sqrtnint(lim+1, k), 3, listput(s, m^k - 1))); Set(s); }
list(lim) = setminus(list2(lim), list1(lim));

Crossrefs

Complement of the disjoint union of A390155 and A390156 within A045542.

Cf. A001597, A073010, A353238, A389764, A389765.

Sequence in context: A316792 A358974 A110864 * A046102 A307793 A145542

Adjacent sequences: A390154 A390155 A390156 * A390158 A390159 A390160

Keyword

nonn,easy

Author

Amiram Eldar, Oct 27 2025

Status

approved