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A337076 Niven numbers (A005349) with a record gap to the next Niven number.

%I A337076 #6 Aug 14 2020 08:08:11
%S A337076 1,10,12,63,72,90,288,378,558,2889,3784,6480,19872,28971,38772,297864,
%T A337076 478764,589860,989867,2879865,9898956,49989744,88996914,689988915,
%U A337076 879987906,989888823,2998895823,6998899824,8889999624,8988988866,9879997824,18879988824,286889989806
%N A337076 Niven numbers (A005349) with a record gap to the next Niven number.
%C A337076 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 A337076 Kennedy and Cooper (1984) proved that the asymptotic density of the Niven numbers is 0. Therefore, this sequence is infinite.
%C A337076 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 A337076 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 A337076 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 A337076 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 A337076 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 A337076 Cf. A005349, A330927, A337077.
%K A337076 nonn,base
%O A337076 1,2
%A A337076 _Amiram Eldar_, Aug 14 2020