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

36, 40, 72, 96, 126, 176, 200, 225, 234, 252, 280, 297, 320, 408, 520, 550, 576, 588, 600, 648, 690, 714, 735, 736, 768, 780, 784, 816, 850, 855, 896, 945, 972, 1026, 1040, 1064, 1092, 1160, 1188, 1216, 1242, 1248, 1275, 1280, 1296, 1300, 1350, 1404, 1530

Offset

1,1

Comments

First differs from A060292 at n = 4.

If neither of the two unordered triples contains a 1, then the corresponding term is also in A337080. - Matej Veselovac, Sep 14 2020

Links

Alois P. Heinz, Table of n, a(n) for n = 1..10000

L. F. Meyers and R. See, The Census-Taker Problem, Math. Mag. 63 (1990) 86-88

I. J. L. Garces and M. L. Loyola, Revisiting a Number-Theoretic Puzzle: The Census-Taker Problem, Intersection, 11 (2010), pp. 28-38; arXiv:1204.2071 [math.HO], 2012.

Example

36 = 6*6*1 = 9*2*2. 6+6+1 = 9+2+2. So 36 is in the sequence.
This example also explains the name of the sequence:
"A census taker knocks on a door. A mother answers.
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."
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."
What are the ages of the children?" (Garces and Loyola 2010, 28)
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.

Maple

b:= proc(n, k, t) option remember; expand(`if`(t=1, `if`(k<n, 0, x^n),
     add(`if`(d>k, 0, b(n/d, d, t-1)*x^d), d=numtheory[divisors](n))))
    end:
q:= n-> (p-> degree(p)=2 and coeff(p, x, 2)=1)(
             add(x^i, i=[coeffs(b(n$2, 3))])):
select(q, [$1..2000])[];  # Alois P. Heinz, May 16 2020

Mathematica

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]}]];
q[n_] := Exponent[#, x]==2 && Coefficient[#, x, 2]==1& @ Total[ x^CoefficientList[b[n, n, 3], x]];
Select[Range[2000], q] (* Jean-François Alcover, Nov 16 2020, after Alois P. Heinz *)

Crossrefs

Subset of A060292 (at least two unordered triples of positive numbers have product n and equal sums).

Cf. A337080 (at least two unordered factorizations of n have equal sums of factors).

Sequence in context: A276710 A181484 A060292 * A305942 A261265 A344808

Adjacent sequences: A334908 A334909 A334910 * A334912 A334913 A334914

Keyword

nonn

Author

Thomas Gawlick, May 16 2020

Status

approved