A216765 - OEIS

A216765

Perfect powers (squares, cubes, etc., not including 1) plus 1.

5, 9, 10, 17, 26, 28, 33, 37, 50, 65, 82, 101, 122, 126, 129, 145, 170, 197, 217, 226, 244, 257, 290, 325, 344, 362, 401, 442, 485, 513, 530, 577, 626, 677, 730, 785, 842, 901, 962, 1001, 1025, 1090, 1157, 1226, 1297, 1332, 1370, 1445, 1522, 1601, 1682, 1729, 1765

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Integers of the form m^k + 1 for integers m, k >= 2.

REFERENCES

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.6, p. 113.

LINKS

Table of n, a(n) for n=1..53.

Eugène Catalan, Note sur la sommation de quelques séries, Journal de Mathématiques Pures et Appliquées, Serie 1, Volume 7 (1842), pp. 1-12. See p. 8, eq. (24).

Vsevolod F. Lev, Re: The sequence of 'perfect' powers, Number Theory List, May 21, 2002.

FORMULA

a(n) = A001597(n+1) + 1 = A045542(n) + 2. [corrected by Georg Fischer, Jun 21 2020]

Sum_{n>=1} 1/a(n) = Pi^2/3 - 5/2 (Catalan, 1842; Lev, 2002). - Amiram Eldar, Oct 15 2020, Oct 14 2025

EXAMPLE

a(1) = 2^2 + 1; a(2) = 2^3 + 1; a(3) = 3^2 + 1; a(4) = 2^4 + 1.

MATHEMATICA

seq[lim_] := Union[Table[m^k + 1, {k, 2, Log2[lim + 1]}, {m, 2, Surd[lim + 1, k]}] // Flatten]; seq[2000] (* Amiram Eldar, Oct 14 2025 *)

PROG

(Python)

from sympy import mobius, integer_nthroot

from oeis_sequences.OEISsequences import bisection

def A216765(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 21 2025

CROSSREFS

Cf. A001597, A045542, A175064, A195055, A214390, A214391.

Sequence in context: A136318 A242139 A393025 * A117771 A227853 A275240

Adjacent sequences: A216762 A216763 A216764 * A216766 A216767 A216768

KEYWORD

nonn,easy

AUTHOR

Jonathan Vos Post, Sep 15 2012

STATUS

approved