A385084 Smallest integer value of Manhattan(v) + Euclidean(v) where v is a vector of n positive integers, and Manhattan(v) = Sum v[i] and Euclidean(v) = sqrt(Sum v[i]^2).

2, 12, 8, 6, 12, 10, 14, 14, 12, 16, 20, 22, 18, 22, 26, 20, 24, 28, 26, 30, 30, 28, 36, 34, 30, 38, 36, 36, 42, 38, 42, 44, 40, 44, 46, 42, 48, 48, 54, 50, 50, 56, 52, 58, 58, 54, 60, 60, 56, 62, 66, 64, 64, 68, 66, 66, 70, 68, 74, 74, 70, 76, 76, 72, 78, 78, 80

Offset: 1

Author

Charles L. Hohn, Jun 17 2025

Keywords

• nonn

Comments

Manhattan and Euclidean are distance measures from the origin for v as coordinates of a point in n-dimensional space.

Integer Euclidean(v) requires that v is a Pythagorean n-tuple.

These distances are a walk from the origin by a positive integer number of unit steps in each of the n dimensions, and a return to the origin by a straight line which is also an integer number of unit steps.

When n is a square, a(n) = n + sqrt(n) from v all 1's.

Examples

			For n=2, v is the legs of a Pythagorean triangle and the smallest Manhattan + Euclidean is a(2) = 12 from v = {3,4}.
a(3) = 8 because 1 + 2 + 2 + sqrt(1^2 + 2^2 + 2^2) =  1 + 2 + 2 + 3 = 8.
a(4) = 6 because 1 + 1 + 1 + 1 + sqrt(1^2 + 1^2 + 1^2 + 1^2) = 1 + 1 + 1 + 1 + 2 = 6.
a(29) = 42: 3^2 + (4 * 2^2) + (24 * 1^2) = 7^2, and 6^2 + (28 * 1^2) = 8^2 (first term with multiple different examples).
		

Programs

PARI

a(n)={my(r=oo); for(c=0, oo, if(n+sqrtint(n-1)+1+c>=r, return(r)); forpart(p=c, my(s=sum(i=1, #p, (p[i]+1)^2)+(n-#p)); if(issquare(s), r=min(r, n+c+sqrtint(s))), , n))}