A344375 Numbers n that can be written as the concatenation ab such that n mod (a*b) = a+b.

23, 29, 33, 39, 43, 49, 53, 59, 63, 69, 73, 79, 83, 89, 93, 99, 103, 109, 113, 119, 123, 129, 133, 139, 143, 149, 153, 159, 163, 169, 173, 179, 183, 189, 193, 199, 203, 209, 211, 213, 219, 223, 229, 233, 239, 243, 249, 253, 259, 263, 269, 273, 279, 283, 289, 293, 299, 303, 309, 311, 313, 319, 323

Offset: 1

Author

J. M. Bergot and Robert Israel, May 16 2021

Keywords

• nonn
• base

Comments

Leading zeros are not allowed in b.

Includes all n >= 23 with n == 3 or 9 (mod 10), and all n >= 211 with n == 11 (mod 100).

Are there terms not of these forms?

Includes forms n == 142857 (mod 1000000) for n >= 2142857. - Michael S. Branicky, May 16 2021

Examples

			a(25) = 143 is a term because 143 mod (14*3) = 17 = 14+3.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A268044.

Programs

Maple

filter:= proc(n) local k,a,b;
  for k from 1 to ilog10(n) do
    a:= n mod 10^k;
    b:= (n-a)/10^k;
    if a < 10^(k-1) then next fi;
    if n mod (a*b) = a+b then return true fi
  od;
  false
end proc:
select(filter, [$10..1000]);

Python

def ok(n):
  s = str(n)
  for i in range(1, len(s)):
    if s[i] == '0': continue
    a, b = int(s[:i]), int(s[i:])
    if n%(a*b) == a+b: return True
  return False
print(list(filter(ok, range(1, 324)))) # Michael S. Branicky, May 16 2021