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%I #27 Jan 05 2025 19:51:41
%S 1,2,3,4,5,6,131052,491424,1275140,1310412,1474224,1614623,1912700,
%T 2031132,2142014,2457024,2550260,3229223,3931224,4422624,4914024,
%U 5405424,5654912,5920222,7013180,7125325,7371024,8073023,8347710,9424832,10000095,10000096,10000097
%N Starts of runs of 5 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="/A330928/b330928.txt">Table of n, a(n) for n = 1..10000</a>
%H 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 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 Wikipedia, <a href="https://en.wikipedia.org/wiki/Harshad_number">Harshad number</a>.
%H 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.
%F This A330928 = { A005349(k) | A005349(k+4) = A005349(k)+4 }. - _M. F. Hasler_, Jan 03 2022
%e 131052 is a term since 131052 is divisible by 1 + 3 + 1 + 0 + 5 + 2 = 12, 131053 is divisible by 13, 131054 is divisible by 14, 131055 is divisible by 15, and 131056 is divisible by 16.
%t nivenQ[n_] := Divisible[n, Total @ IntegerDigits[n]]; niv = nivenQ /@ Range[5]; seq = {}; Do[niv = Join[Rest[niv], {nivenQ[k]}]; If[And @@ niv, AppendTo[seq, k - 4]], {k, 5, 10^7}]; seq
%t SequencePosition[Table[If[Divisible[n,Total[IntegerDigits[n]]],1,0],{n,10^7+200}],{1,1,1,1,1}][[;;,1]] (* _Harvey P. Dale_, Dec 24 2023 *)
%o (Magma) f:=func<n|n mod &+Intseq(n) eq 0>; a:=[]; for k in [1..11000000] do if forall{m:m in [0..4]|f(k+m)} then Append(~a,k); end if; end for; a; // _Marius A. Burtea_, Jan 03 2020
%o (PARI) {first( N=50, LEN=5, L=List())= for(n=1,oo, n+=LEN; for(m=1,LEN, n--%sumdigits(n) && next(2)); listput(L,n); N--|| break);L} \\ _M. F. Hasler_, Jan 03 2022
%Y Cf. A005349, A060159; A330927, A154701, A141769, A330929, A330930 (same for 2, 3, 4, 6, 7 consecutive harshad numbers).
%K nonn,base,changed
%O 1,2
%A _Amiram Eldar_, Jan 03 2020