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.

1, 0, 6, 18, 28, 24, 48, 60, 84, 120, 120, 120, 180, 180, 240, 360, 360, 360, 360, 672, 720, 720, 720, 840, 840, 1080, 1260, 1260, 1260, 1680, 1680, 1680, 2160, 2520, 2520, 2520, 2520, 2520, 2520, 3360, 4320, 5040, 5040, 5040, 5040, 5040, 5040, 5040, 5040

Offset

1,3

Comments

This sequence is the generalization of the problem A1737 proposed on French mathematical site Diophante (see link).

a(2) = 0 but all other terms are nonzero.

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?

Links

Table of n, a(n) for n=1..49.

Diophante, A1737 - Fidèles au rendez-vous (in French).

Example

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):
.
   n    m   d_1 d_2 d_3 d_4 d_5 d_6 d_7 d_8 d_9 d10 d11 d12
   -----------------------------------------------------------
   1    1    1
   2    0
   3    6    1   2   3
   4   18    1   2   6   9
   5   28    1   2   4   7  14
   6   24    1   2   3   4   6   8
   7   48    1   2   3   4   8  16  24
   8   60    1   2   3   4   5  10  15  20
   9   84    1   2   3   4   6   7  12  21  28
  10  120    1   2   3   4   5   6  15  20  24  40
  11  120    1   2   3   4   5   6   8  12  15  24  40
  12  120    1   2   3   4   5   6   8  10  12  15  24  30
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.

Prog

(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)); ); }
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

Crossrefs

Cf. A000396, A081512.

Sequence in context: A337324 A140450 A157800 * A352061 A225110 A077660

Adjacent sequences: A355225 A355226 A355227 * A355229 A355230 A355231

Keyword

nonn

Author

Bernard Schott, Jun 25 2022

Extensions

More terms from Jinyuan Wang, Jun 25 2022

Status

approved