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

1,2

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.

Example

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

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 *)

Prog

(Magma) f:=func<n|n mod &+Intseq(n) eq 0>; 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
(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

Crossrefs

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

Sequence in context: A290951 A114800 A079170 * A319387 A076105 A094280

Adjacent sequences: A330924 A330925 A330926 * A330928 A330929 A330930

Keyword

nonn,base

Author

Amiram Eldar, Jan 03 2020

Status

approved