A308534 - OEIS

%I #48 Sep 07 2019 06:09:40

%S 126,129,134,136,138,147,148,156,158,159,162,163,165,167,169,174,176,

%T 178,183,185,187,189,192,195,196,198,219,235,236,239,249,253,256,258,

%U 263,265,267,268,269,276,279,285,289,291,293,294,296,297,298,318,326,329,345,346,347,348,349,354,356,358,364,365,367,368,374,376

%N Numbers k such that the LCM of all possible sums of two digits of k (at distinct positions) is strictly greater than k.

%C The only integer k for which the LCM of the sum of any two digits (at distinct positions) is equal to k is 32760. [Edited by _M. F. Hasler_, Jun 06 2019]

%C The sequence contains 1359845 terms, the largest term is 12251988. - _David A. Corneth_, Jun 06 2019

%C Between 10^k and 10^(k+1) there are (241, 3002, 27238, 183258, 1053624, 92482) terms, for k = 2..7. Any term must have at least two digits larger than 1, and must not have more than one digit 0. The smallest term with a digit 0 is 409, the largest one is 12250988. The number of terms in [10^k,10^(k+1)] with a digit 9 is (111, 1571, 16856, 153986, 1053624, 92482) (increasing from 50% to 100% for > 6 digits, for an overall 97% of terms having a digit 9), and with a digit 0: (15, 525, 6133, 51911, 344730, 53837) (increasing from 6% to about 50%), again for k = 2..7. - _M. F. Hasler_, Jun 07 2019

%C From _Bernard Schott_, Aug 08 2019: (Start)

%C A proof that this sequence is finite.

%C Suppose there exists a term m >= 10^9, so with d >= 9 digits, that belongs to this sequence. With these d digits of m, it is possible to get binomial(d,2) = d*(d-1)/2 >= 9*8/2 = 36 sums of integers with two digits of m. Now, the only possible sums obtained with two digits of m go from 0+1=1 to 9+9=18 and the LCM of {1, 2, ..., 18} = A003418(18) = 12252240. Therefore, the LCM of all possible sums of two digits for each number m is <= 12252240, and here m >= 10^9. We have these successive inequalities: LCM of all possible sums of two digits for number m <= 12252240 < 10^9 <= m; but, with the name, the LCM of all possible sums of two digits of m must be > m. There is a contradiction, there is no term m >= 10^9 in the sequence and thus this sequence is finite.

%C We remark that the largest term 12251988 is < A003418(18) = 12252240. (End)

%H Jean-Marc Falcoz, <a href="/A308534/b308534.txt">Table of n, a(n) for n = 1..30482</a>

%e a(1) = 126; the three sums of two digits of 126 are {3, 7, 8} with LCM = 168;

%e a(2) = 129; the three sums of two digits of 129 are {3, 10, 11} with LCM = 330;

%e a(3) = 134; the three sums of two digits of 134 are {4, 5, 7} with LCM = 140;

%e etc.

%o (PARI) is(n) = my(d = digits(n), l = List()); for(i = 1, #d - 1, for(j = i+1, #d, listput(l, d[i]+d[j]))); lcm(Set(l)) > n \\ _David A. Corneth_, Jun 06 2019

%Y Cf. A003418 (LCM).

%K base,fini,nonn

%O 1,1

%A _Eric Angelini_ and _Jean-Marc Falcoz_, Jun 06 2019