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A060292 At least two unordered triples of positive numbers have product n and equal sums.

%I A060292 #18 May 30 2020 09:19:29
%S A060292 36,40,72,90,96,126,144,168,176,200,225,234,240,252,270,280,288,297,
%T A060292 320,360,396,408,420,432,450,480,504,520,540,546,550,560,576,588,600,
%U A060292 630,648,672,675,690,714,720,735,736,768,780,784,800,816,840,850,855
%N A060292 At least two unordered triples of positive numbers have product n and equal sums.
%H A060292 Alois P. Heinz, <a href="/A060292/b060292.txt">Table of n, a(n) for n = 1..10000</a> (first 160 terms from Carmine Suriano)
%e A060292 36=6*6*1=9*2*2. 6+6+1=9+2+2. so 36 is in the sequence.
%p A060292 N:= 1000: # to get all entries <= N
%p A060292 for i from 1 to N do R[i]:= {} od:
%p A060292 A:= {}:
%p A060292 for a from 1 to floor(N^(1/3)) do
%p A060292   for b from a to floor((N/a)^(1/2)) do
%p A060292     for c from b to floor(N/(a*b)) do
%p A060292        p:= a*b*c;
%p A060292        s:= a+b+c;
%p A060292        if member(s,R[p]) then A:= A union {p}
%p A060292        else R[p]:= R[p] union {s}
%p A060292        fi;
%p A060292 od od od:
%p A060292 A;
%p A060292 # if using Maple 11 or earlier, uncomment the next line
%p A060292 # sort(convert(A,list)); # _Robert Israel_, Feb 09 2015
%p A060292 # second Maple program:
%p A060292 b:= proc(n, k, t) option remember; expand(`if`(t=0, `if`(k<n, 0, x^n),
%p A060292      add(`if`(d>k, 0, b(n/d, d, t-1)*x^d), d=numtheory[divisors](n))))
%p A060292     end:
%p A060292 a:= proc(n) option remember; local k; for k from 1+
%p A060292       `if`(n=1, 0, a(n-1)) while max(coeffs(b(k$2, 2)))<2 do od; k
%p A060292     end:
%p A060292 seq(a(n), n=1..50);  # _Alois P. Heinz_, May 16 2020
%t A060292 b[n_, k_, t_] := b[n, k, t] = Expand[If[t == 0, If[k < n, 0, x^n], Sum[If[d > k, 0, b[n/d, d, t - 1] x^d], {d, Divisors[n]}]]];
%t A060292 a[n_] := a[n] = Module[{k}, For[k = 1 + If[n == 1, 0, a[n - 1]], Max[ CoefficientList[b[k, k, 2], x]] < 2, k++]; k];
%t A060292 Array[a, 52] (* _Jean-François Alcover_, May 30 2020, after _Alois P. Heinz_ *)
%Y A060292 Cf. A060275.
%K A060292 easy,nonn
%O A060292 1,1
%A A060292 _Naohiro Nomoto_, Mar 24 2001
%E A060292 Name changed by _Robert Israel_, Feb 09 2015