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

Author

Amiram Eldar, May 05 2020

Keywords

• nonn
• base

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.

Examples

			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).
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Shyam Sunder Gupta, Smith Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 4, 127-157.
• 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.

Crossrefs

Intersection of A005349 and A006753.

Programs

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]

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