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Numbers divisible by the LCM of all sums of any two digits (at distinct positions).
1

%I #17 Jan 14 2021 21:17:24

%S 1,2,3,4,5,6,7,8,9,10,12,18,20,21,24,27,30,36,40,42,45,48,50,54,60,63,

%T 70,72,80,81,84,90,102,110,120,140,150,204,210,220,240,252,280,306,

%U 330,360,408,420,440,480,510,540,550,630,660,770,840,880,990,1110,1140,1212,1320,1440,1512,2016

%N Numbers divisible by the LCM of all sums of any two digits (at distinct positions).

%C The sequence trivially includes the single-digit numbers, their multiples of 10, and more generally any near-repdigit number having all digits equal and nonzero, except for a trailing 0, i.e., of the form d*(10^n-10)/9, 1 <= d <= 9, n > 1.

%C If a number has more than one digit 0, the relevant LCM is zero and will never divide such a number.

%C Between 84 and 10^4 there are only 13 terms without digit 0: (252, 1212, 1512, 2112, 2424, 2772, 3636, 4224, 4848, 6216, 6336, 8316, 8448), cf. A308561.

%e For single-digit numbers 1, 2, ..., 9, there is no sum of two digits, and since the LCM of an empty set is taken to be 1, it divides the number.

%e Two-digit terms are in the sequence iff they are divisible by the sum of their digits. This includes multiples of ten: 10, 20, ..., 90, but also less trivial examples as 12, 18, 21, ...

%e The first 3-digit term having no digit 0 is 252, divisible by both, 2 + 5 = 7 and 2 + 2 = 4.

%o (PARI) is(n,d=digits(n))={n<10||vecsort(d)[2]&&0==n%lcm(concat(vector(-1+n=#d,i,t=d[1+n-i];vector(n-i,j,t+d[j]))))}

%Y Cf. A308534, A308561.

%K nonn,base

%O 1,2

%A _M. F. Hasler_, Jun 07 2019