A362023 a(n) is the least positive integer whose decimal expansion is the concatenation of the decimal expansions of two numbers whose product is n.

11, 12, 13, 14, 15, 16, 17, 18, 19, 25, 111, 26, 113, 27, 35, 28, 117, 29, 119, 45, 37, 112, 123, 38, 55, 126, 39, 47, 129, 56, 131, 48, 113, 134, 57, 49, 137, 138, 133, 58, 141, 67, 143, 114, 59, 146, 147, 68, 77, 105, 151, 134, 153, 69, 115, 78, 157, 158

Offset: 1

Author

Rémy Sigrist, Apr 05 2023

Keywords

• nonn
• base

Comments

For any prime number p, a(p) is the least of the concatenation of p with 1 or the concatenation of 1 with p.

Examples

			The first terms, alongside an appropriate way to split them into two factors, are:
  n   a(n)  a(n)
  --  ----  ----
   1    11   1*1
   2    12   1*2
   3    13   1*3
   4    14   1*4
   5    15   1*5
   6    16   1*6
   7    17   1*7
   8    18   1*8
   9    19   1*9
  10    25   2*5
  11   111  11*1
  12    26   2*6
  13   113  1*13
  14    27   2*7
  15    35   3*5
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A055642, A067574, A347471, A362022 (binary variant).

Programs

Mathematica

Table[Min@ Map[FromDigits[Join @@ #] &, Join @@ {#, Reverse /@ #}] &@ Map[IntegerDigits[#] &, Transpose@{#, n/#}, {2}] &@ TakeWhile[Divisors[n], # <= Sqrt[n] &], {n, 60}] (* Michael De Vlieger, Apr 07 2023 *)

PARI

a(n, base = 10) = { my (v = oo); fordiv (n, d, v = min(v, n/d * base^#digits(d, base) + d);); return (v); }

Python

from sympy import divisors
def a(n): return min(int(str(d)+str(n//d)) for d in divisors(n))
print([a(n) for n in range(1, 61)]) # Michael S. Branicky, Apr 05 2023

Formula

a(n) <= 10*n + 1.

a(n) <= 10^A055642(n) + n.

a(n) = Min_{d | n} A067574(n/d, d).