A147763 a(n) = smallest k such that n = d*(d-b)*(d+c), where b, c, d >= 0 and k = b+c.

0, 1, 2, 1, 4, 2, 6, 0, 2, 4, 10, 1, 12, 6, 4, 2, 16, 1, 18, 3, 6, 10, 22, 2, 4, 12, 0, 5, 28, 3, 30, 2, 10, 16, 6, 1, 36, 18, 12, 3, 40, 5, 42, 9, 2, 22, 46, 1, 6, 3, 16, 11, 52, 3, 10, 5, 18, 28, 58, 2, 60, 30, 4, 0, 12, 9, 66, 15, 22, 5, 70, 3, 72, 36, 2, 17, 10, 11, 78, 1, 6, 40, 82, 4, 16

Offset: 1

Author

Samuel Zbarsky (sa_zbarsky(AT)yahoo.com), Nov 11 2008

Keywords

• nonn
• easy

Examples

			n = 9 = 3*1*3 = 3*(3-2)*(3+0) with d = 3, b = 2, c = 0. So a(9) = k = 2+0 = 2, since there is no solution with k = 1.
		

Links

• David A. Corneth, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A056737.

Programs

Magma

[ Min(v) where v is [ b+c: d, y, z in Divisors(n) | d*(d-b)*(d+c) eq n and d ge 0 and b ge 0 and c ge 0 where b is d-y where c is z-d ]: n in [1..85] ]; // Klaus Brockhaus, Nov 19 2008

PARI

first(n) = {my(res = vector(n, i, i), t = 0); for(i = 1, n, for(j = i, n \ i, for(k = j, n \ (i * j), res[i * j * k] = min(res[i * j * k], k - i)))); res} \\ David A. Corneth, May 12 2018

Formula

a(n) = n-1 for n prime; a(n) = 0 for n a 3rd power. - Klaus Brockhaus, Nov 19 2008

Extensions

Definition and example edited, and extended beyond a(42) by Klaus Brockhaus, Nov 19 2008