A391414 - OEIS

%I #17 Dec 17 2025 15:57:22

%S 1,3,16,184,2696,47466,979776,23059204,567108864,14712104501,

%T 421504185344,13218256749601,450353989316608,16565151205544957,

%U 654244800082329600,27614800115689879553,1240529732459024678912,59095217374989483261925,2975557672677668838178816,157904201452248753415276001

%N a(n) is the median of the set of the distinct values of (n-1)^n, (n-1)^(n+1), n^(n-1), n^(n+1), (n+1)^(n-1), (n+1)^n.

%C The median of 6 sorted values x(i), i=1..6, is defined as (x(3) + x(4))/2.

%F a(n) = ((n-1)^n + (n+1)^n)/2 = A062024(n) for n >= 9. - _Robert Israel_, Dec 16 2025

%e a(1) = 1: 0^1, 0^2, 1^0, 1^2, 2^0, 2^1 -> median {0, 1, 2} = 1;

%e a(2) = 3: 1^2, 1^3, 2^1, 2^3, 3^1, 3^2 -> median {1, 2, 3, 8, 9} = 3;

%e a(3) = 16: 2^3, 2^4, 3^2, 3^4, 4^2, 4^3 -> median {8, 9, 16, 64, 81} = 16 (=A051489(2)/2);

%e a(4) = 184: 3^4, 3^5, 4^3, 4^5, 5^3, 5^4 -> median {64, 81, 125, 243, 625, 1024} = (125 + 243)/2 = 184 (=A051489(3)/2);

%e a(5) = 2696 (=A051489(4)/2);

%e a(6) = 47466 (=A051489(5)/2);

%e a(7) = 979776: 6^7, 6^8, 7^6, 7^8, 8^6, 8^7 -> median {117649, 262144, 279936, 1679616, 2097152, 5764801} = (279936 + 1679616)/2 = 979776, whereas A051489(6)/2 = 970880.

%t a[n_]:=Median[Union[{(n-1)^n,(n-1)^(n+1),n^(n-1),n^(n+1),(n+1)^(n-1),(n+1)^n}]];Array[a,20] (* _James C. McMahon_, Dec 16 2025 *)

%o (Python)

%o def A391414(n): return ((d:=sorted({a:=(n-1)**n,a*(n-1),b:=n**(n-1),b*n**2,c:=(n+1)**(n-1),c*(n+1)}))[(l:=len(d))-1>>1]+d[l>>1])>>1 # _Chai Wah Wu_, Dec 17 2025

%Y Cf. A051489, A062024.

%K nonn

%O 1,2

%A _Hugo Pfoertner_, Dec 16 2025