A334527 Numbers that are both Niven numbers and Smith numbers.

4, 27, 378, 576, 588, 645, 648, 666, 690, 825, 915, 1872, 1908, 1962, 2265, 2286, 2484, 2556, 2688, 2934, 2970, 3168, 3258, 3294, 3345, 3366, 3390, 3564, 3615, 3690, 3852, 3864, 3930, 4428, 4464, 4557, 4880, 5526, 6084, 6315, 7695, 8154, 8736, 8883, 9015, 9036

Offset

1,1

Comments

McDaniel (1990) proved that there exist infinitely many numbers which are both base-b Niven numbers and base-b Smith numbers, for all bases b >= 8.

Links

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

Wayne L. McDaniel, On the Intersection of the Sets of Base b Smith Numbers and Niven Numbers, Missouri Journal of Mathematical Sciences, Vol. 2, No. 3 (1990), pp. 132-136.

Example

27 is a term since it is a Niven number (2 + 7 = 9 is a divisor of 27) and a Smith number (27 = 3 * 3 * 3 and 2 + 7 = 3 + 3 + 3).

Mathematica

digSum[n_] := Plus @@ IntegerDigits[n]; nivenSmithQ[n_] := Divisible[n, (ds = digSum[n])] && CompositeQ[n] && Plus @@ (Last@# * digSum[First@#] & /@ FactorInteger[n]) == ds; Select[Range[10^4], nivenSmithQ]

Prog

(Python)
from sympy import factorint
def sd(n): return sum(map(int, str(n)))
def ok(n):
  sdn = sd(n)
  if sdn == 0 or n%sdn != 0: return False # not Niven
  f = factorint(n)
  return sum(f[p] for p in f) > 1 and sdn == sum(sd(p)*f[p] for p in f)
print(list(filter(ok, range(9999)))) # Michael S. Branicky, Apr 27 2021

Crossrefs

Intersection of A005349 and A006753.

Sequence in context: A320961 A239726 A058155 * A104169 A359904 A271615

Adjacent sequences: A334524 A334525 A334526 * A334528 A334529 A334530

Keyword

nonn,base

Author

Amiram Eldar, May 05 2020

Status

approved