A330927 Numbers k such that both k and k + 1 are Niven numbers.

1, 2, 3, 4, 5, 6, 7, 8, 9, 20, 80, 110, 111, 132, 152, 200, 209, 224, 399, 407, 440, 480, 510, 511, 512, 629, 644, 735, 800, 803, 935, 999, 1010, 1011, 1014, 1015, 1016, 1100, 1140, 1160, 1232, 1274, 1304, 1386, 1416, 1455, 1520, 1547, 1651, 1679, 1728, 1853

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

			1 is a term since 1 and 1 + 1 = 2 are both Niven numbers.
		

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.

Crossrefs

Cf. A005349, A060159, A141769, A154701, A328205, A328209, A328213, A330713, A330928, A330929, A330930, A330931.

Programs

Magma

f:=func; a:=[]; for k in [1..2000] do  if forall{m:m in [0..1]|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]]; nq1 = nivenQ[1]; seq = {}; Do[nq2 = nivenQ[k]; If[nq1 && nq2, AppendTo[seq, k - 1]]; nq1 = nq2, {k, 2, 2000}]; seq
SequencePosition[Table[If[Divisible[n,Total[IntegerDigits[n]]],1,0],{n,2000}],{1,1}][[;;,1]] (* Harvey P. Dale, Dec 24 2023 *)

Python

from itertools import count, islice
def agen(): # generator of terms
    h1, h2 = 1, 2
    while True:
        if h2 - h1 == 1: yield h1
        h1, h2 = h2, next(k for k in count(h2+1) if k%sum(map(int, str(k))) == 0)
print(list(islice(agen(), 52))) # Michael S. Branicky, Mar 17 2024