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A308534
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Numbers k such that the LCM of all possible sums of two digits of k (at distinct positions) is strictly greater than k.
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3
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126, 129, 134, 136, 138, 147, 148, 156, 158, 159, 162, 163, 165, 167, 169, 174, 176, 178, 183, 185, 187, 189, 192, 195, 196, 198, 219, 235, 236, 239, 249, 253, 256, 258, 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
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OFFSET
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1,1
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COMMENTS
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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]
The sequence contains 1359845 terms, the largest term is 12251988. - David A. Corneth, Jun 06 2019
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
A proof that this sequence is finite.
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.
We remark that the largest term 12251988 is < A003418(18) = 12252240. (End)
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LINKS
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EXAMPLE
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a(1) = 126; the three sums of two digits of 126 are {3, 7, 8} with LCM = 168;
a(2) = 129; the three sums of two digits of 129 are {3, 10, 11} with LCM = 330;
a(3) = 134; the three sums of two digits of 134 are {4, 5, 7} with LCM = 140;
etc.
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PROG
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(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
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CROSSREFS
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KEYWORD
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base,fini,nonn
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AUTHOR
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STATUS
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approved
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