A125112 Numbers which are not the sum of 3 nonzero squares, but which can be expressed as the product of two numbers that are the sum of 3 nonzero squares.

63, 87, 135, 156, 159, 183, 207, 231, 252, 279, 303, 319, 327, 348, 351, 375, 399, 423, 444, 447, 471, 476, 495, 519, 540, 543, 551, 567, 572, 583, 591, 615, 624, 636, 639, 663, 671, 687, 700, 711, 732, 735, 759, 783, 807, 828, 831, 847, 855, 879, 903, 924

Offset: 1

Author

Artur Jasinski, Nov 21 2006

Keywords

• nonn

Comments

Intersection of A004214 with products of pairs of terms of A000408.

Examples

			a(2) = 87 = 3 * 29 = (1^2+1^2+1^2) * (4^2+3^2+2^2)
87 does not have a partition as a sum x^2+y^2+z^2 with x,y,z>0
63=3*21; 87=3*29; 135=3*45; 156=6*26; 572=22*26;
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000408 (sums of 3 nonzero squares), A004214 (not sums of 3 nonzero squares).

Programs

Maple

isA000408 := proc(n) local a,b,c2 ; a:=1; while a^2A125112 := proc(n) local d,i; if isA000408(n) then RETURN(false) ; else d := numtheory[divisors](n) ; for i from 1 to nops(d) do if isA000408(op(i,d)) and isA000408(n/op(i,d)) then RETURN(true) ; fi ; od ; RETURN(false) ; fi ; end: for an from 1 to 1600 do if isA125112(an) then printf("%d,",an) ; fi ; od ; # R. J. Mathar, Nov 23 2006

Mathematica

isA000408[n_] := Module[{a, b, c2}, a = 1; While[a^2 < n, b = 1; While[b <= a && a^2 + b^2 < n, c2 = n - a^2 - b^2; If[IntegerQ@Sqrt@c2, Return[True]]; b++]; a++]; Return[False]];
isA125112[n_] := Module[{d, i}, If[isA000408[n], Return[False], d = Divisors[n]; For[i = 1, i <= Length[d], i++, If[isA000408[d[[i]]] && isA000408[n/d[[i]]], Return[True]]]; Return[False]]];
Select[Range[1600], isA125112] (* Jean-François Alcover, Jul 22 2024, after R. J. Mathar *)

Extensions

Edited and extended by R. J. Mathar and Ray Chandler, Nov 23 2006