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Starts of runs of 6 consecutive Niven (or Harshad) numbers (A005349).
9

%I #19 Mar 17 2024 07:33:26

%S 1,2,3,4,5,10000095,10000096,12751220,14250624,22314620,22604423,

%T 25502420,28501224,35521222,41441420,41441421,51004820,56511023,

%U 57002424,70131620,71042422,71253024,97740760,102009620,111573020,114004824,121136420,124324220,124324221

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

%C Cooper and Kennedy proved that there are infinitely many runs of 20 consecutive Niven numbers. Therefore this sequence is infinite.

%D Jean-Marie De Koninck, Those Fascinating Numbers, American Mathematical Society, 2009, p. 36, entry 110.

%H Amiram Eldar, <a href="/A330929/b330929.txt">Table of n, a(n) for n = 1..4000</a>

%H Curtis Cooper and Robert E. Kennedy, <a href="http://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 Helen G. Grundman, <a href="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 Wikipedia, <a href="https://en.wikipedia.org/wiki/Harshad_number">Harshad number</a>.

%H Brad Wilson, <a href="http://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 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 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 (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 Cf. A005349, A060159, A141769, A154701, A330927, A330928, A330930.

%K nonn,base

%O 1,2

%A _Amiram Eldar_, Jan 03 2020