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.
1, 3, 27, 15, 63, 135, 384, 315, 960, 1995, 1155, 1575, 2835, 3840, 5775, 4095, 6720, 14400, 14175, 10395, 13440, 20475, 20160, 36855, 48384, 26880, 46080, 108675, 57600, 51975, 40320, 190575, 100800, 193536, 107520, 172800, 126720, 80640, 174720, 120960, 744975
Offset: 0
Keywords
Examples
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.
Links
- Project Euler, Problem 135: Same differences
- Project Euler, Problem 136: Singleton difference
Crossrefs
Cf. A334566.
Programs
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Maple
g:= proc(y,m) local d; d:= m/(4*y)+y/4; d::posint and y > d end proc: f:= proc(m) local L; nops(select(g, numtheory:-divisors(m),m)); end proc: V:= Array(0..50): count:= 0: for x from 1 while count < 51 do v:= f(x); if v <= 50 and V[v] = 0 then V[v]:= x; count:= count+1; fi od: convert(V,list); # Robert Israel, May 19 2020
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Mathematica
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 *)
Extensions
More terms from Giovanni Resta, May 19 2020
Comments