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A330929 Starts of runs of 6 consecutive Niven (or Harshad) numbers (A005349).

%I A330929 #21 Jan 05 2025 19:51:41
%S A330929 1,2,3,4,5,10000095,10000096,12751220,14250624,22314620,22604423,
%T A330929 25502420,28501224,35521222,41441420,41441421,51004820,56511023,
%U A330929 57002424,70131620,71042422,71253024,97740760,102009620,111573020,114004824,121136420,124324220,124324221
%N A330929 Starts of runs of 6 consecutive Niven (or Harshad) numbers (A005349).
%C A330929 Cooper and Kennedy proved that there are infinitely many runs of 20 consecutive Niven numbers. Therefore this sequence is infinite.
%D A330929 Jean-Marie De Koninck, Those Fascinating Numbers, American Mathematical Society, 2009, p. 36, entry 110.
%H A330929 Amiram Eldar, <a href="/A330929/b330929.txt">Table of n, a(n) for n = 1..4000</a>
%H A330929 Curtis Cooper and Robert E. Kennedy, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/31-2/cooper.pdf">On consecutive Niven numbers</a>, Fibonacci Quarterly, Vol. 21, No. 2 (1993), pp. 146-151.
%H A330929 Helen G. Grundman, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/32-2/grundman.pdf">Sequences of consecutive Niven numbers</a>, Fibonacci Quarterly, Vol. 32, No. 2 (1994), pp. 174-175.
%H A330929 Wikipedia, <a href="https://en.wikipedia.org/wiki/Harshad_number">Harshad number</a>.
%H A330929 Brad Wilson, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/35-2/wilson.pdf">Construction of 2n consecutive n-Niven numbers</a>, Fibonacci Quarterly, Vol. 35, No. 2 (1997), pp. 122-128.
%e A330929 10000095 is a term since 10000095 is divisible by 1 + 0 + 0 + 0 + 0 + 0 + 9 + 5 = 15, 10000096 is divisible by 16, ..., and 10000100 is divisible by 2.
%t A330929 nivenQ[n_] := Divisible[n, Total @ IntegerDigits[n]]; niv = nivenQ /@ Range[6]; seq = {}; Do[niv = Join[Rest[niv], {nivenQ[k]}]; If[And @@ niv, AppendTo[seq, k - 5]], {k, 6, 10^7}]; seq
%o A330929 (Magma) f:=func<n|n mod &+Intseq(n) eq 0>; a:=[]; for k in [1..30000000] do  if forall{m:m in [0..5]|f(k+m)} then Append(~a,k); end if; end for; a; // _Marius A. Burtea_, Jan 03 2020
%Y A330929 Cf. A005349, A060159, A141769, A154701, A330927, A330928, A330930.
%K A330929 nonn,base
%O A330929 1,2
%A A330929 _Amiram Eldar_, Jan 03 2020