A104166 Repdigit Smith numbers.

4, 22, 666, 1111, 6666666, 4444444444, 44444444444444444444, 555555555555555555555555555, 55555555555555555555555555555555, 4444444444444444444444444444444444444444444444444444444

Offset: 1

Author

Shyam Sunder Gupta, Mar 10 2005

Keywords

• base
• nonn

Links

• Shyam Sunder Gupta, Smith Numbers.
• Shyam Sunder Gupta, Smith Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 4, 127-157.

Crossrefs

Cf. A006753.

Subsequence of both A098834 and A104171.

Programs

Mathematica

d[n_]:=IntegerDigits[n]; tr[n_]:=Transpose[FactorInteger[n]]; a[n_]:=NestList[FromDigits[Flatten[d[{#,n}]]]&,n,55]; t={}; Do[If[!PrimeQ[n]&&Total[d[n]]==Total[d@tr[n][[1]]*tr[n][[2]],2],AppendTo[t,n]],{n,Drop[Union[Flatten[Table[a[k],{k,9}]]],1]}]; t (* Jayanta Basu, Jun 04 2013 *)

Python

from sympy import factorint
from itertools import product
def sd(n): return sum(map(int, str(n)))
def smith(n):
  f = factorint(n)
  return sum(f[p] for p in f) > 1 and sd(n) == sum(sd(p)*f[p] for p in f)
def repsto(limit):
  yield from range(min(limit, 9)+1)
  for rep in range(2, 10**len(str(limit))):
    for digit in "123456789":
      out = int(digit*rep)
      if out > limit: return
      yield out
print(list(filter(smith, repsto(10**32)))) # Michael S. Branicky, Apr 22 2021