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%I #6 Aug 14 2020 08:08:11
%S 1,10,12,63,72,90,288,378,558,2889,3784,6480,19872,28971,38772,297864,
%T 478764,589860,989867,2879865,9898956,49989744,88996914,689988915,
%U 879987906,989888823,2998895823,6998899824,8889999624,8988988866,9879997824,18879988824,286889989806
%N Niven numbers (A005349) with a record gap to the next Niven number.
%C The corresponding record gaps are 1, 2, 6, 7, 8, 10, 12, 14, 18, 23, 32, 36, 44, 45, 54, 60, 66, 72, 88, 90, 99, 108, 126, 135, 144, 150, 153, 192, 201, 234, 258, 276, 294, ...
%C Kennedy and Cooper (1984) proved that the asymptotic density of the Niven numbers is 0. Therefore, this sequence is infinite.
%C De Koninck and Doyon proved that for sufficiently large k the least number m such that the interval[m, m+k-1] does not contain any Niven numbers is < (100*(k+2))^(k+3).
%H Jean-Marie De Koninck and Nicolas Doyon, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL6/Doyon/doyon.html">Large and Small Gaps Between Consecutive Niven Numbers</a>, J. Integer Seqs., Vol. 6, 2003, Article 03.2.5.
%H R. E. Kennedy and C. N. Cooper, <a href="http://www.jstor.org/stable/2686395">On the natural density of the Niven numbers</a>, The College Mathematics Journal, Vol. 15, No. 4 (1984), pp. 309-312.
%e 10 is a term since it is a Niven number, and the next Niven number is 12, with a gap 12 - 10 = 2, which is a record, since all the numbers below 10 are also Niven numbers.
%t nivenQ[n_] := Divisible[n, Plus @@ IntegerDigits[n]]; gapmax = 0; n1 = 1; s = {}; Do[If[nivenQ[n], gap = n - n1; If[gap > gapmax, gapmax = gap; AppendTo[s, n1]]; n1 = n], {n, 2, 10^6}]; s
%Y Cf. A005349, A330927, A337077.
%K nonn,base
%O 1,2
%A _Amiram Eldar_, Aug 14 2020
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