A186129 - OEIS

%I #17 Sep 08 2022 08:45:55

%S 18,27,32,36,45,48,50,54,63,64,72,75,80,81,90,96,98,99,100,108,112,

%T 117,125,126,128,135,144,147,150,153,160,162,171,175,176,180,189,192,

%U 196,198,200,207,208,216,224,225,234,240,242,243,245,250,252,256,261

%N 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.

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

%C 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.

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

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

%H Klaus Brockhaus, <a href="/A186129/b186129.txt">Table of n, a(n) for n = 1..1178</a> (terms <= 5000)

%e 18 = 2+6+2+8; for k=2 we have 2+2 = 6-2 = 2*2 = 8/2 = 4, hence 18 is a term.

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

%o (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

%Y Cf. A000041, A013929, A046790, A072903.

%K nonn

%O 1,1

%A _Manuel Valdivia_, Feb 13 2011