A329547 - OEIS

%I #51 Sep 18 2024 16:33:10

%S 1,2,2,3,3,4,4,5,6,7,7,8,8,9,9,10,10,11,11,12,12,13,13,14,15,16,16,17,

%T 17,18,18,19,19,20,20,21,21,22,22,23,23,24,24,25,25,26,26,27,28,29,29,

%U 30,30,31,31,32,32,33,33,34,34,35,35,36,36,37,37,38,38,39,39,40,40

%N Number of natural numbers k <= n such that k^k is a square.

%C 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.

%C It seems the unrepeated terms form A266304 \ {0}. - _Ivan N. Ianakiev_, Nov 21 2019

%C Indices of unrepeated terms are A081349. - _Rémy Sigrist_, Dec 07 2019

%H David A. Corneth, <a href="/A329547/b329547.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = floor(n/2) + ceiling(floor(sqrt(n))/2).

%e a(5) = 3 because among 1^1, 2^2, ..., 5^5 there are 3 squares: 1^1, 2^2, and 4^4.

%t Table[Floor[n/2] + Ceiling[Floor[Sqrt[n]]/2], {n, 1, 100}]

%o (PARI) a(n) = sum(k=1, n, issquare(k^k)); \\ _Michel Marcus_, Nov 17 2019

%o (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

%o (PARI) a(n) = n\2 + (sqrtint(n)+1)\2 \\ _David A. Corneth_, Dec 07 2019

%o (Python)

%o from math import isqrt

%o def A329547(n): return (n>>1)+(isqrt(n)+1>>1) # _Chai Wah Wu_, Sep 18 2024

%Y Cf. A081349, A176693, A266304.

%K nonn,easy

%O 1,2

%A _Pablo Hueso Merino_, Nov 16 2019