Consecutive Harshad Numbers
H.G. Grundman proved in 1994 that, in base 10, no 21 consecutive integers are all Harshad numbers. She also found the smallest 20 consecutive integers that are all Harshad numbers; they exceed 1044363342786.
In binary, there are infinitely many sequences of four consecutive Harshad numbers; in ternary, there are infinitely many sequences of six consecutive Harshad numbers. Both of these facts were proven by T. Cai in 1996.
In general, such maximal sequences run from N · bk - b to N · bk + (b-1), where b is the base, k is a relatively large power, and N is a constant. Interpolating zeroes into N will not change the sequence of digital sums, so it is possible to convert any solution into a larger one by interpolating a suitable number of zeroes, just as 21 and 201 and 2001 are all Harshad numbers base 10. Thus any solution implies an infinite class of solutions.
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