%I #19 Sep 26 2020 11:07:53
%S 4,8,12,16,20,24,28,32,36,40,44,48,52,56,60,64,68,72,76,80,84,88,90,
%T 92,96,100,104,108,112,116,120,124,128,132,136,140,144,148,152,156,
%U 160,164,168,172,176,180,184,188,192,196,200,204,208,212,216,220,224,228,232,236,240,244,248
%N Complement of A337037.
%C Numbers with a pair of unordered factorizations whose sums of factors are the same.
%C All terms of the sequence are composite.
%C The smallest odd term of the sequence is a(174) = 675. This is a term of the sequence because 675 = 27*5*5 = 9*3*25 and 27+5+5 = 9+3+25 = 37.
%C Terms of the sequence are used in variations of a logic puzzle known as "Ages of Three Children Puzzle" or "Census-taker problem". For the original puzzle, see A334911.
%C If a number m is in the sequence, then all multiples of m are in the sequence. For example, multiples of 4 are in the sequence because there always exist at least two factorizations 4*k = 2*2*k whose factors sum to the same value 4+k = 2+2+k.
%C Numbers m such that A069016(m) < A001055(m). - _Michel Marcus_, Aug 15 2020
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/UnorderedFactorization.html">Unordered Factorization</a>.
%e All unordered factorization of 90 are 90 = 45*2 = 30*3 = 18*5 = 15*6 = 15*3*2 = 10*9 = 9*5*2 = 10*3*3 = 6*5*3 = 5*3*3*2. Corresponding sums of factors are not all distinct: 90, 57, 33, 23, 21, 20, 19, 16, 16, 14, 13 because the sum 16 = 10+3+3 = 9+5+2 appears twice. Therefore 90 is in the sequence.
%e All unordered factorization of 30 are 30 = 15*2 = 10*3 = 6*5 = 5*3*2. Corresponding sums of factors are all distinct: 30 = 30, 17 = 15+2, 13 = 10+3, 11 = 6+5, 10 = 2+3+5. Therefore 30 is not in the sequence.
%o (PARI) factz(n, minn) = {my(v=[]); fordiv(n, d, if ((d>=minn) && (d<=sqrtint(n)), w = factz(n/d, d); for (i=1, #w, w[i] = concat([d], w[i]);); v = concat(v, w););); concat(v, [[n]]);}
%o factorz(n) = factz(n, 2);
%o isok(n) = my(vs = apply(x->vecsum(x), factorz(n))); #vs != #Set(vs); \\ _Michel Marcus_, Aug 14 2020
%Y Cf. A334911 (census-taker numbers).
%Y Cf. A337037 (complement), A337081.
%Y Cf. A001055 (number of unordered factorizations of n), A074206 (number of ordered factorizations of n).
%Y Cf. A056472 (all factorizations of n), A069016 (number of distinct sums).
%K nonn,easy
%O 1,1
%A _Matej Veselovac_, Aug 14 2020
%E Edited by _N. J. A. Sloane_, Sep 14 2020