A128201 Union of positive squares and the odd numbers.

1, 3, 4, 5, 7, 9, 11, 13, 15, 16, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 36, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 64, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 100, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123

Offset: 1

Author

Reinhard Zumkeller, Mar 04 2007

Keywords

• nonn
• easy

Comments

Range of A128200.

Positive numbers n such that n^((1 + n)/2) is an integer. - Gionata Neri, May 07 2016

Links

• R. Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Partial sums given by A157130. - Gerald Hillier, Feb 25 2009

See A176693 for the union of even numbers and the squares. - M. F. Hasler, Apr 19 2015

Cf. A157502, A157512.

Programs

Mathematica

f[n_] := Block[{s = Range[n]^2, t}, Union[s, Range[1, Last@ s, 2]] // Sort]; f@ 12 (* Michael De Vlieger, Apr 16 2015 *)

PARI

A128201(n)=!(bittest(n=2*n-round(sqrt(2*n)),0)||issquare(n))+n \\ Based on Hiliers's formula. - M. F. Hasler, Apr 19 2015

PARI

is_A128201(n)=bittest(n,0)||issquare(n) \\ M. F. Hasler, Apr 19 2015

Python

from math import isqrt
def A128201(n):
    def f(x): return n+(x>>1)-(isqrt(x)>>1)
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Oct 02 2024

Formula

a(n) = f(n,1,1,2), where f(n,i,m,x) = if i=n then m; else if m+1=x^2 then f(n,i+1,m+1,x); else if m+1>x^2 then f(n,i+1,m+1,x+2); else f(n,i+1,m+2,x).

Set R(n) = 2*n - round(sqrt(2*n)); then a(n) = R(n) + sign(frac(sqrt(R(n)))) * (not(R(n) mod 2)). - Gerald Hillier, Apr 16 2015