A334567 - OEIS

%I #21 May 20 2020 15:34:53

%S 1,3,27,15,63,135,384,315,960,1995,1155,1575,2835,3840,5775,4095,6720,

%T 14400,14175,10395,13440,20475,20160,36855,48384,26880,46080,108675,

%U 57600,51975,40320,190575,100800,193536,107520,172800,126720,80640,174720,120960,744975

%N Least value m > 0 such that Diophantine equation z^2 - y^2 - x^2 = m, when the positive integers, x, y and z are consecutive terms of an arithmetic progression, has exactly n solutions.

%C Equivalently, if d is the common difference of the arithmetic progression (x, y, z), then a(n) is the smallest integer m such that the Diophantine equation y * (4d-y) = m with y>0, d>0 and y-d >0 has exactly n solutions (see A334566).

%C This sequence is not increasing: a(2) = 27 > a(3) = 15.

%H Project Euler, <a href="https://projecteuler.net/problem=135">Problem 135: Same differences</a>

%H Euler project, <a href="https://projecteuler.net/problem=136">Problem 136: Singleton difference</a>

%e a(4) = 63 because 11^2-7^2-3^2 = 13^2-9^2-5^2 = 27^2-21^2-15^2 = 79^2-63^2-47^2 = 63 and there is no term m < 63 in the context such that z^2 - y^2 - x^2 = m has 4 solutions.

%p g:= proc(y,m) local d;

%p   d:= m/(4*y)+y/4;

%p   d::posint and y > d

%p end proc:

%p f:= proc(m) local L;

%p   nops(select(g, numtheory:-divisors(m),m));

%p end proc:

%p V:= Array(0..50): count:= 0:

%p for x from 1 while count < 51 do

%p   v:= f(x);

%p   if v <= 50 and V[v] = 0 then V[v]:= x; count:= count+1;

%p   fi

%p od:

%p convert(V,list); # _Robert Israel_, May 19 2020

%t ok[n_, x_] := Block[{d = (x + n/x)/4}, IntegerQ[d] && x > d]; t = Table[ Length@ Select[ Divisors[n], ok[n, #] &], {n, 21000}]; k=0; Reap[ While[ (v = Position[ t, k++]) != {}, Sow[v[[1, 1]]]]][[2, 1]] (* _Giovanni Resta_, May 19 2020 *)

%Y Cf. A334566.

%K nonn

%O 0,2

%A _Bernard Schott_, May 19 2020

%E More terms from _Giovanni Resta_, May 19 2020