A375724 a(n) is the first Smith number with at least n digits.

4, 22, 121, 1086, 10086, 100066, 1000165, 10000426, 100000165, 1000000165, 10000000165, 100000000498, 1000000000066, 10000000000615, 100000000000786, 1000000000000426, 10000000000000246, 100000000000000642, 1000000000000000462, 10000000000000000246, 100000000000000000282, 1000000000000000000966

Offset

1,1

Comments

a(n) is the least composite k >= 10^(n-1) such that the sum of the decimal digits of k is equal to the sum of the decimal digits of the prime factors of k, counted with multiplicity.

Almost certainly a(n) has exactly n digits, but "at least" is included in the Name since we have no proof of that.

Links

Table of n, a(n) for n=1..22.

Example

a(5) = 10086 because 10086 has digit-sum 15 and 10086 = 2 * 3 * 41^2 with 2 + 3 + (4 + 1) + (4 + 1) = 15, and no k from 10000 to 10085 works.

Maple

f:= proc(n) local t, x;
    for x from 10^(n-1) do
      if isprime(x) then next fi;
      if convert(convert(x, base, 10), `+`) = add(t[2]*convert(convert(t[1], base, 10), `+`), t = ifactors(x)[2]) then return x fi;
    od
end proc:
map(f, [$1..30]);

Prog

(Python)
from sympy import factorint
from itertools import count
def sd(n): return sum(map(int, str(n)))
def is_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 a(n): return next(k for k in count(10**(n-1)) if is_smith(k))
print([a(n) for n in range(1, 23)]) # Michael S. Branicky, Aug 25 2024

Crossrefs

Cf. A006753.

Sequence in context: A098834 A065983 A236576 * A185858 A180034 A260346

Adjacent sequences: A375721 A375722 A375723 * A375725 A375726 A375727

Keyword

nonn,base

Author

Robert Israel, Aug 25 2024

Status

approved