A330928 Starts of runs of 5 consecutive Niven (or harshad) numbers (A005349).

1, 2, 3, 4, 5, 6, 131052, 491424, 1275140, 1310412, 1474224, 1614623, 1912700, 2031132, 2142014, 2457024, 2550260, 3229223, 3931224, 4422624, 4914024, 5405424, 5654912, 5920222, 7013180, 7125325, 7371024, 8073023, 8347710, 9424832, 10000095, 10000096, 10000097

Offset

1,2

Comments

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

References

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

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Curtis Cooper and Robert E. Kennedy, On consecutive Niven numbers, Fibonacci Quarterly, Vol. 21, No. 2 (1993), pp. 146-151.

Helen G. Grundman, Sequences of consecutive Niven numbers, Fibonacci Quarterly, Vol. 32, No. 2 (1994), pp. 174-175.

Wikipedia, Harshad number.

Brad Wilson, Construction of 2n consecutive n-Niven numbers, Fibonacci Quarterly, Vol. 35, No. 2 (1997), pp. 122-128.

Formula

This A330928 = { A005349(k) | A005349(k+4) = A005349(k)+4 }. - M. F. Hasler, Jan 03 2022

Example

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.

Mathematica

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
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 *)

Prog

(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
(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

Crossrefs

Cf. A005349, A060159; A330927, A154701, A141769, A330929, A330930 (same for 2, 3, 4, 6, 7 consecutive harshad numbers).

Sequence in context: A004901 A004912 A292125 * A164699 A228050 A331561

Adjacent sequences: A330925 A330926 A330927 * A330929 A330930 A330931

Keyword

nonn,base

Author

Amiram Eldar, Jan 03 2020

Status

approved