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A334567 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.

%I A334567 #23 May 10 2025 16:39:47
%S A334567 1,3,27,15,63,135,384,315,960,1995,1155,1575,2835,3840,5775,4095,6720,
%T A334567 14400,14175,10395,13440,20475,20160,36855,48384,26880,46080,108675,
%U A334567 57600,51975,40320,190575,100800,193536,107520,172800,126720,80640,174720,120960,744975
%N A334567 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 A334567 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 A334567 This sequence is not increasing: a(2) = 27 > a(3) = 15.
%H A334567 Project Euler, <a href="https://projecteuler.net/problem=135">Problem 135: Same differences</a>
%H A334567 Project Euler, <a href="https://projecteuler.net/problem=136">Problem 136: Singleton difference</a>
%e A334567 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 A334567 g:= proc(y,m) local d;
%p A334567   d:= m/(4*y)+y/4;
%p A334567   d::posint and y > d
%p A334567 end proc:
%p A334567 f:= proc(m) local L;
%p A334567   nops(select(g, numtheory:-divisors(m),m));
%p A334567 end proc:
%p A334567 V:= Array(0..50): count:= 0:
%p A334567 for x from 1 while count < 51 do
%p A334567   v:= f(x);
%p A334567   if v <= 50 and V[v] = 0 then V[v]:= x; count:= count+1;
%p A334567   fi
%p A334567 od:
%p A334567 convert(V,list); # _Robert Israel_, May 19 2020
%t A334567 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 A334567 Cf. A334566.
%K A334567 nonn
%O A334567 0,2
%A A334567 _Bernard Schott_, May 19 2020
%E A334567 More terms from _Giovanni Resta_, May 19 2020