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

Author

Reinhard Zumkeller, Apr 14 2011

Keywords

• nonn

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

Crossrefs

Union of A000290 and A000079.

Disjoint union of A000290 and A004171.

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

Programs

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

Mathematica

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

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

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