A186129 Numbers that can be partitioned into four parts s, t, u, v such that s+k = t-k = u*k = v/k for some k > 1.

18, 27, 32, 36, 45, 48, 50, 54, 63, 64, 72, 75, 80, 81, 90, 96, 98, 99, 100, 108, 112, 117, 125, 126, 128, 135, 144, 147, 150, 153, 160, 162, 171, 175, 176, 180, 189, 192, 196, 198, 200, 207, 208, 216, 224, 225, 234, 240, 242, 243, 245, 250, 252, 256, 261

Offset: 1

Author

Manuel Valdivia, Feb 13 2011

Keywords

• nonn

Comments

Equivalently, solutions n to a*(b+1)^2 = b*n with a > b >= 2.

The general rule to obtain such a partition is to start with any number b > 1 and one of its multiples a = k*b (k > 1 and a < n) and let s = a-b, t = a+b, u = a/b and v = a*b.

Sequence appears to be a subsequence of A013929, of A046790, and of A072903.

Examples

			18 = 2+6+2+8; for k=2 we have 2+2 = 6-2 = 2*2 = 8/2 = 4, hence 18 is a term.
45 = 8+12+5+20; for k=2 we have 8+2 = 12-2 = 5*2 = 20/2 = 10, hence 45 is a term.
		

References

• José Estalella, Ciencia Recreativa. Gustavo Gili - Editor. Barcelona, 1918, pp. 5-6.

Links

• Klaus Brockhaus, Table of n, a(n) for n = 1..1178 (terms <= 5000)

Crossrefs

Cf. A000041, A013929, A046790, A072903.

Programs

Magma

[ n: n in [1..300] | exists{ b: b in [2..n] | exists{ a: a in [b+1..n div 4] | n*b eq a*(b+1)^2 } } ]; // Klaus Brockhaus, Feb 15 2011