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%I #16 Mar 17 2024 07:33:37
%S 623,846,2358,4206,4878,6127,6222,6223,12438,16974,21006,27070,31295,
%T 33102,33103,35343,37134,37630,37638,40703,43263,45550,48190,49230,
%U 52590,53262,53263,56110,59630,66198,66702,66703,67878,69310,69487,72655,74766,77230,77958
%N Starts of runs of 3 consecutive Niven numbers in base 2 (A049445).
%C Cai proved that there are infinitely many runs of 4 consecutive Niven numbers in base 2. Therefore this sequence is infinite.
%D József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 4, p. 382.
%H Amiram Eldar, <a href="/A330932/b330932.txt">Table of n, a(n) for n = 1..10000</a>
%H Tianxin Cai, <a href="https://www.fq.math.ca/Scanned/34-2/cai1.pdf">On 2-Niven numbers and 3-Niven numbers</a>, Fibonacci Quarterly, Vol. 34, No. 2 (1996), pp. 118-120.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Harshad_number#Consecutive_harshad_numbers">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 623 is a term since 623, 624 and 625 are all Niven numbers in base 2.
%t binNivenQ[n_] := Divisible[n, Total @ IntegerDigits[n, 2]]; bin = binNivenQ /@ Range[3]; seq = {}; Do[bin = Join[Rest[bin], {binNivenQ[k]}]; If[And @@ bin, AppendTo[seq, k - 2]], {k, 3, 8*10^4}]; seq
%o (Magma) f:=func<n|n mod &+Intseq(n,2) eq 0>; a:=[]; for k in [1..80000] do if forall{m:m in [0..2]|f(k+m)} then Append(~a,k); end if; end for; a; // _Marius A. Burtea_, Jan 03 2020
%Y Cf. A049445, A154701, A328210, A328214, A330931, A330933.
%K nonn,base
%O 1,1
%A _Amiram Eldar_, Jan 03 2020
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