OFFSET
1,2
COMMENTS
For even k, k^k is always a square. For odd k, k^k is a square if and only if k is a square.
It seems the unrepeated terms form A266304 \ {0}. - Ivan N. Ianakiev, Nov 21 2019
Indices of unrepeated terms are A081349. - Rémy Sigrist, Dec 07 2019
LINKS
David A. Corneth, Table of n, a(n) for n = 1..10000
FORMULA
a(n) = floor(n/2) + ceiling(floor(sqrt(n))/2).
EXAMPLE
a(5) = 3 because among 1^1, 2^2, ..., 5^5 there are 3 squares: 1^1, 2^2, and 4^4.
MATHEMATICA
Table[Floor[n/2] + Ceiling[Floor[Sqrt[n]]/2], {n, 1, 100}]
PROG
(PARI) a(n) = sum(k=1, n, issquare(k^k)); \\ Michel Marcus, Nov 17 2019
(PARI) first(n) = my(res=vector(n), inc); res[1] = 1; for(i=2, n, inc = (1-(i%2)) || issquare(i); res[i] = res[i-1] + inc); res \\ David A. Corneth, Dec 07 2019
(PARI) a(n) = n\2 + (sqrtint(n)+1)\2 \\ David A. Corneth, Dec 07 2019
(Python)
from math import isqrt
def A329547(n): return (n>>1)+(isqrt(n)+1>>1) # Chai Wah Wu, Sep 18 2024
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Pablo Hueso Merino, Nov 16 2019
STATUS
approved