A374735 a(n) is the least k > 0 such that n and k*n can be added without carries in decimal.

1, 1, 1, 1, 1, 2, 2, 3, 5, 10, 1, 1, 1, 1, 1, 2, 2, 3, 10, 20, 1, 1, 1, 1, 1, 2, 2, 6, 5, 30, 1, 1, 1, 1, 1, 4, 7, 3, 20, 40, 1, 1, 1, 1, 1, 10, 5, 3, 5, 50, 2, 2, 2, 2, 6, 2, 2, 6, 30, 60, 2, 2, 2, 2, 5, 2, 2, 3, 15, 70, 3, 3, 3, 7, 3, 4, 12, 13, 40, 80, 5, 5

Offset: 0

Author

Rémy Sigrist, Jul 18 2024

Keywords

• nonn
• base
• easy

Examples

			For n = 8:
- 1*8 = 8; computing 8 + 8 requires a carry,
- 2*8 = 16; computing 8 + 16 requires a carry,
- 3*8 = 24; computing 8 + 24 requires a carry,
- 4*8 = 32; computing 8 + 32 requires a carry,
- 5*8 = 40; computing 8 + 40 does not require a carry,
- so a(8) = 5.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A007091, A261891 (analog for binary), A353623 (analog for balanced ternary), A374736.

Programs

PARI

a(n, base = 10) = { for (k = 1, oo, if (sumdigits((k+1)*n, base) == sumdigits(n, base) + sumdigits(k*n, base), return (k););); }

Python

from itertools import count
def A374735(n):
    s = list(map(int,str(n)[::-1]))
    return next(k for k in count(1) if all(a+b<=9 for a, b in zip(s,map(int,str(k*n)[::-1])))) # Chai Wah Wu, Jul 19 2024

Formula

a(n) = 1 iff n belongs to A007091.

a(10*n) = a(n).