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}, 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
KEYWORD
nonn,easy
AUTHOR
Reinhard Zumkeller, Mar 04 2007
EXTENSIONS
Mathematica program modified by Harvey P. Dale, May 02 2026
STATUS
approved