A391414 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.

1, 3, 16, 184, 2696, 47466, 979776, 23059204, 567108864, 14712104501, 421504185344, 13218256749601, 450353989316608, 16565151205544957, 654244800082329600, 27614800115689879553, 1240529732459024678912, 59095217374989483261925, 2975557672677668838178816, 157904201452248753415276001

Offset

1,2

Comments

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

Links

Table of n, a(n) for n=1..20.

Formula

a(n) = ((n-1)^n + (n+1)^n)/2 = A062024(n) for n >= 9. - Robert Israel, Dec 16 2025

Example

a(1) = 1: 0^1, 0^2, 1^0, 1^2, 2^0, 2^1 -> median {0, 1, 2} = 1;
a(2) = 3: 1^2, 1^3, 2^1, 2^3, 3^1, 3^2 -> median {1, 2, 3, 8, 9} = 3;
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);
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);
a(5) = 2696 (=A051489(4)/2);
a(6) = 47466 (=A051489(5)/2);
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.

Mathematica

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 *)

Prog

(Python)
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

Crossrefs

Cf. A051489, A062024.

Sequence in context: A152554 A254430 A180721 * A192668 A045990 A007006

Adjacent sequences: A391411 A391412 A391413 * A391415 A391416 A391417

Keyword

nonn

Author

Hugo Pfoertner, Dec 16 2025

Status

approved