A188915 Union of squares and powers of 2.

0, 1, 2, 4, 8, 9, 16, 25, 32, 36, 49, 64, 81, 100, 121, 128, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 512, 529, 576, 625, 676, 729, 784, 841, 900, 961, 1024, 1089, 1156, 1225, 1296, 1369, 1444, 1521, 1600, 1681, 1764, 1849, 1936, 2025, 2048, 2116, 2209, 2304, 2401, 2500, 2601, 2704, 2809, 2916

Offset

0,3

Comments

A188916 and A188917 give positions where squares and powers of 2 occur:

n^2: a(A188916(n)) = A000290(n);

2^n: a(A188917(n)) = A000079(n);

4^n: a(A006127(n)) = A000302(n), A006127 is the intersection of A188916 and A188917.

Links

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Chai Wah Wu, Algorithms for Complementary Sequences, Integers (2025) Vol. 25, Art. No. A95. See p. 21.

Formula

A010052(a(n)) + A209229(a(n)) > 0. - Reinhard Zumkeller, May 19 2015

Sum_{n>=1} 1/a(n) = Pi^2/6 + 2/3. - Amiram Eldar, Apr 13 2025

Mathematica

seq[lim_] := Union[2^Range[1, Floor[Log2[lim]], 2], Range[0, Floor[Sqrt[lim]]]^2]; seq[3000] (* Amiram Eldar, Apr 13 2025 *)

Prog

(Haskell)
import Data.List.Ordered (union)
a188915 n = a188915_list !! n
a188915_list = union a000290_list a000079_list
-- Reinhard Zumkeller, May 19 2015, Apr 14 2011
(Python)
from math import isqrt
def A188915(n):
    def bisection(f, kmin=0, kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-isqrt(x)-((m:=x.bit_length()-1)>>1)-(m&1)
    return bisection(f, n-1, n**2) # Chai Wah Wu, Sep 19 2024

Crossrefs

Union of A000290 and A000079.

Disjoint union of A000290 and A004171.

Cf. A000302, A010052, A209229, A006127, A188916, A188917.

Sequence in context: A049439 A251642 A079931 * A055008 A204826 A241010

Adjacent sequences: A188912 A188913 A188914 * A188916 A188917 A188918

Keyword

nonn

Author

Reinhard Zumkeller, Apr 14 2011

Status

approved