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A355228 a(n) is the smallest integer m such that there exist n of its distinct divisors (d_1, d_2, ..., d_n) with the property that m = d_1 + d_2 + ... + d_n = lcm(d_1, d_2, ..., d_n), or 0 if no such number m exists.

%I A355228 #37 Jun 27 2022 10:03:21
%S A355228 1,0,6,18,28,24,48,60,84,120,120,120,180,180,240,360,360,360,360,672,
%T A355228 720,720,720,840,840,1080,1260,1260,1260,1680,1680,1680,2160,2520,
%U A355228 2520,2520,2520,2520,2520,3360,4320,5040,5040,5040,5040,5040,5040,5040,5040
%N A355228 a(n) is the smallest integer m such that there exist n of its distinct divisors (d_1, d_2, ..., d_n) with the property that m = d_1 + d_2 + ... + d_n = lcm(d_1, d_2, ..., d_n), or 0 if no such number m exists.
%C A355228 This sequence is the generalization of the problem A1737 proposed on French mathematical site Diophante (see link).
%C A355228 a(2) = 0 but all other terms are nonzero.
%C A355228 a(n) >= A081512(n) because in A081512, it is not required that m = lcm(d_1, d_2, ..., d_n). Currently, the strict inequality happens for n = 4 and n = 5; are there other such cases?
%H A355228 Diophante, <a href="http://www.diophante.fr/problemes-par-themes/arithmetique-et-algebre/a1-pot-pourri/4960-a1737-fideles-au-rendez-vous">A1737 - Fidèles au rendez-vous</a> (in French).
%e A355228 In the following triangle, the n-th row gives an example of a set of n divisors d_1, ..., d_n of a(n) such that a(n) = d_1 + ... + d_n = lcm(d_1, ..., d_n):
%e A355228 .
%e A355228    n    m   d_1 d_2 d_3 d_4 d_5 d_6 d_7 d_8 d_9 d10 d11 d12
%e A355228    -----------------------------------------------------------
%e A355228    1    1    1
%e A355228    2    0
%e A355228    3    6    1   2   3
%e A355228    4   18    1   2   6   9
%e A355228    5   28    1   2   4   7  14
%e A355228    6   24    1   2   3   4   6   8
%e A355228    7   48    1   2   3   4   8  16  24
%e A355228    8   60    1   2   3   4   5  10  15  20
%e A355228    9   84    1   2   3   4   6   7  12  21  28
%e A355228   10  120    1   2   3   4   5   6  15  20  24  40
%e A355228   11  120    1   2   3   4   5   6   8  12  15  24  40
%e A355228   12  120    1   2   3   4   5   6   8  10  12  15  24  30
%e A355228 However, for a given value of a(n) = m, there may be more than one way to choose d_1, ..., d_n. For example, for n=10, a(10)=120 and all seventeen solutions provided by _Jinyuan Wang_ in the Comments section of A081512 are also solutions here.
%o A355228 (PARI) isok(m, n) = {my(d=divisors(m)); if (#d<n, return(0)); forsubset([#d, n], s, my(vd = vector(n, k, d[s[k]])); if (lcm(vd) == vecsum(vd), return(1)););}
%o A355228 a(n) = {if (n==1, return(1)); if (n==2, return(0)); my(m=1); while (!isok(m, n), m++); m;} \\ _Michel Marcus_, Jun 25 2022
%Y A355228 Cf. A000396, A081512.
%K A355228 nonn
%O A355228 1,3
%A A355228 _Bernard Schott_, Jun 25 2022
%E A355228 More terms from _Jinyuan Wang_, Jun 25 2022