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A334911 Census-taker numbers: Numbers k such that exactly two unordered triples of positive numbers have product k and equal sums.

%I A334911 #43 Nov 16 2020 09:29:04
%S A334911 36,40,72,96,126,176,200,225,234,252,280,297,320,408,520,550,576,588,
%T A334911 600,648,690,714,735,736,768,780,784,816,850,855,896,945,972,1026,
%U A334911 1040,1064,1092,1160,1188,1216,1242,1248,1275,1280,1296,1300,1350,1404,1530
%N A334911 Census-taker numbers: Numbers k such that exactly two unordered triples of positive numbers have product k and equal sums.
%C A334911 First differs from A060292 at n = 4.
%C A334911 If neither of the two unordered triples contains a 1, then the corresponding term is also in A337080. - _Matej Veselovac_, Sep 14 2020
%H A334911 Alois P. Heinz, <a href="/A334911/b334911.txt">Table of n, a(n) for n = 1..10000</a>
%H A334911 L. F. Meyers and R. See, <a href="https://www.jstor.org/stable/2691063?seq=1#metadata_info_tab_contents">The Census-Taker Problem</a>, Math. Mag. 63 (1990) 86-88
%H A334911 I. J. L. Garces and M. L. Loyola, <a href="https://arxiv.org/abs/1204.2071">Revisiting a Number-Theoretic Puzzle: The Census-Taker Problem</a>, Intersection, 11 (2010), pp. 28-38; arXiv:1204.2071 [math.HO], 2012.
%e A334911 36 = 6*6*1 = 9*2*2. 6+6+1 = 9+2+2. So 36 is in the sequence.
%e A334911 This example also explains the name of the sequence:
%e A334911 "A census taker knocks on a door. A mother answers.
%e A334911 The census taker says, "I need to know the number of children you have, and their ages." The woman responds in puzzle-ese, "I have three daughters, the product of their ages is 36, and the sum of their ages is equal to the house number next door."
%e A334911 The census taker, who never wastes questions, computes for a while and then asks, "Does your oldest daughter love dogs?" The mother answers affirmatively. The census taker says, "Thank you. I now know the ages."
%e A334911 What are the ages of the children?" (Garces and Loyola 2010, 28)
%e A334911 3264 is not in the sequence: (34, 32, 3) and (48, 17, 4) give sum 69, but (48, 34, 2), (64, 17, 3) and (68, 12, 4) give sum 84.
%p A334911 b:= proc(n, k, t) option remember; expand(`if`(t=1, `if`(k<n, 0, x^n),
%p A334911      add(`if`(d>k, 0, b(n/d, d, t-1)*x^d), d=numtheory[divisors](n))))
%p A334911     end:
%p A334911 q:= n-> (p-> degree(p)=2 and coeff(p, x, 2)=1)(
%p A334911              add(x^i, i=[coeffs(b(n$2, 3))])):
%p A334911 select(q, [$1..2000])[];  # _Alois P. Heinz_, May 16 2020
%t A334911 b[n_, k_, t_] := b[n, k, t] = If[t==1, If[k<n, 0, x^n], Sum[If[d>k, 0, b[n/d, d, t-1] x^d], {d, Divisors[n]}]];
%t A334911 q[n_] := Exponent[#, x]==2 && Coefficient[#, x, 2]==1& @ Total[ x^CoefficientList[b[n, n, 3], x]];
%t A334911 Select[Range[2000], q] (* _Jean-François Alcover_, Nov 16 2020, after _Alois P. Heinz_ *)
%Y A334911 Subset of A060292 (at least two unordered triples of positive numbers have product n and equal sums).
%Y A334911 Cf. A337080 (at least two unordered factorizations of n have equal sums of factors).
%K A334911 nonn
%O A334911 1,1
%A A334911 _Thomas Gawlick_, May 16 2020