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

1, 2, 3, 4, 5, 10000095, 10000096, 12751220, 14250624, 22314620, 22604423, 25502420, 28501224, 35521222, 41441420, 41441421, 51004820, 56511023, 57002424, 70131620, 71042422, 71253024, 97740760, 102009620, 111573020, 114004824, 121136420, 124324220, 124324221

Offset: 1

Author

Amiram Eldar, Jan 03 2020

Keywords

• nonn
• base

Comments

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

Examples

			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.
		

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..4000
• 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.

Crossrefs

Cf. A005349, A060159, A141769, A154701, A330927, A330928, A330930.

Programs

Magma

f:=func; 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

Mathematica

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