A385614 Numbers of the form x^x + y^y, 1 < x < y.

31, 260, 283, 3129, 3152, 3381, 46660, 46683, 46912, 49781, 823547, 823570, 823799, 826668, 870199, 16777220, 16777243, 16777472, 16780341, 16823872, 17600759, 387420493, 387420516, 387420745, 387423614, 387467145, 388244032, 404197705, 10000000004

Offset: 1

Author

Sean A. Irvine, Jul 04 2025

Keywords

• nonn
• easy

Comments

Terms are all combinations of 1 < x < y ordered by increasing y then increasing x, since the largest of one y is strictly less than the smallest of the next: (y-1)^(y-1) + y^y < 2^2 + (y+1)^(y+1) for y >= 3. - Kevin Ryde, Jul 06 2025

Examples

			31 is in the sequence because 31 = 2^2 + 3^3.
		

Links

• Sean A. Irvine, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000312, A173054, A385232.

Programs

PARI

a(n) = my(r,s=sqrtint((n-1)<<1,&r), x=2 + if(r>1, y=3 + s-(rKevin Ryde, Jul 06 2025

Python

from math import isqrt, comb
def A385614(n):
    y = (m:=isqrt(k:=n<<1))+(k>m*(m+1))+2
    x = n-comb(y-2,2)+1
    return x**x+y**y # Chai Wah Wu, Jul 07 2025

Formula

a(n) = x^x + y^y where x=A131818(n+1) and y=A133196(n). - Kevin Ryde, Jul 06 2025