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

1,2

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

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

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 *)

Prog

(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

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.

Sequence in context: A190941 A284752 A161153 * A347301 A233514 A096262

Adjacent sequences: A128198 A128199 A128200 * A128202 A128203 A128204

Keyword

nonn,easy

Author

Reinhard Zumkeller, Mar 04 2007

Status

approved