A285815 Numbers k such that, for any divisor d of k, the digital sum of d divides k.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 18, 20, 21, 24, 27, 30, 36, 40, 54, 60, 63, 72, 81, 90, 108, 120, 162, 180, 216, 243, 270, 324, 360, 486, 540, 648, 810, 972, 1080, 1458, 1620, 1944, 2430, 2916, 3240, 4374, 4860, 5832, 7290, 8748, 9720, 13122, 14580, 17496

Offset

1,2

Comments

All terms are Niven numbers (A005349).

All terms > 1 have a prime divisor < 10.

Is this sequence infinite?

Some families of terms:

- 2*3^k with 0 <= k <= 12,

- 2*3^k*5 with 0 <= k <= 10,

- 2^2*3^k with 0 <= k <= 13,

- 2^2*3^k*5 with 0 <= k <= 22,

- 2^3*3^k with 0 <= k <= 13,

- 2^3*3^k*5 with 0 <= k <= 22,

- 3^k with 0 <= k <= 5.

The first 99 terms are 7-smooth (A002473).

From David A. Corneth, Apr 20 2021: (Start)

Let k be a term. If 11|k then (1+1)=2|k so 22|k. Similarily if 22|k then 44|k. If 44|k then 88|k. If 88|k then 176|k. If 176|k then (1+7+6) = 14|k so lcm(176, 14) = 1232. Repeating this a few times we see k > 10^43.

Can we use this to prove if p|k then p <= 7 where p is a prime and k is a term?

(End)

Links

Rémy Sigrist, Table of n, a(n) for n = 1..99

David A. Corneth, Conjectured full sequence with 108 terms. All 9973-smooth terms <= 10^30.

Example

The divisors of 243 are: 1, 3, 9, 27, 81, 243; their digital sums are: 1, 3, 9, 9, 9, 9, all divisors of 243; hence 243 is in the sequence.
14 divides 42, but its digital sum, 5, does not divide 42; hence 42 is not in the sequence.

Prog

(PARI) is(n) = fordiv(n, d, if (n % sumdigits(d), return (0))); return (1)
(Python)
from sympy import divisors
from sympy.ntheory.factor_ import digits
def ok(n):
   return all(n%sum(digits(d)[1:])==0 for d in divisors(n))
print([n for n in range(1, 20001) if ok(n)]) # Indranil Ghosh, Apr 28 2017

Crossrefs

Cf. A002473, A005349.

Sequence in context: A085135 A085133 A308560 * A337741 A110806 A357769

Adjacent sequences: A285812 A285813 A285814 * A285816 A285817 A285818

Keyword

nonn,base

Author

Rémy Sigrist, Apr 27 2017

Status

approved