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Sum-product number
A sum-product number in a given number base is a natural number that is equal to the product of the sum of its digits and the product of its digits.
There are a finite number of sum-product numbers in any given base .[1] In base 10, there are exactly four sum-product numbers (sequence A038369 in the OEIS): 0, 1, 135, and 144.[2]
Definition
Let be a natural number. We define the sum-product function for base to be the following:
where is the number of digits in the number in base , and
is the value of each digit of the number. A natural number is a sum-product number if it is a fixed point for , which occurs if . The natural numbers 0 and 1 are trivial sum-product numbers for all , and all other sum-product numbers are nontrivial sum-product numbers.
For example, the number 144 in base 10 is a sum-product number, because , , and .
A natural number is a sociable sum-product number if it is a periodic point for , where for a positive integer , and forms a cycle of period . A sum-product number is a sociable sum-product number with , and a amicable sum-product number is a sociable sum-product number with .
All natural numbers are preperiodic points for , regardless of the base. This is because for any given digit count , the minimum possible value of is and the maximum possible value of is . The maximum possible digit sum is therefore and the maximum possible digit product is . Thus, the sum-product function value is . This suggests that , or dividing both sides by , . Since , this means that there will be a maximum value where , because of the exponential nature of and the linearity of . Beyond this value , always. Thus, there are a finite number of sum-product numbers,[1] and any natural number is guaranteed to reach a periodic point or a fixed point less than , making it a preperiodic point.
The number of iterations needed for to reach a fixed point is the sum-product function's persistence of , and undefined if it never reaches a fixed point.
Any integer shown to be a sum-product number in a given base must, by definition, also be a Harshad number in that base.
Sum-product numbers and cycles of Fb for specific b
All numbers are represented in base .
Base | Nontrivial sum-product numbers | Cycles |
---|---|---|
2 | (none) | (none) |
3 | (none) | 2 → 11 → 2, 22 → 121 → 22 |
4 | 12 | (none) |
5 | 341 | 22 → 31 → 22 |
6 | (none) | (none) |
7 | 22, 242, 1254, 2343, 116655, 346236, 424644 | |
8 | (none) | |
9 | 13, 281876, 724856, 7487248 | 53 → 143 → 116 → 53 |
10 | 135, 144 | |
11 | 253, 419, 2189, 7634, 82974 | |
12 | 128, 173, 353 | |
13 | 435, A644, 268956 | |
14 | 328, 544, 818C | |
15 | 2585 | |
16 | 14 | |
17 | 33, 3B2, 3993, 3E1E, C34D, C8A2 | |
18 | 175, 2D2, 4B2 | |
19 | 873, B1E, 24A8, EAH1, 1A78A, 6EC4B7 | |
20 | 1D3, 14C9C, 22DCCG | |
21 | 1CC69 | |
22 | 24, 366C, 6L1E, 4796G | |
23 | 7D2, J92, 25EH6 | |
24 | 33DC | |
25 | 15, BD75, 1BBN8A | |
26 | 81M, JN44, 2C88G, EH888 | |
27 | ||
28 | 15B | |
30 | 976, 85MDA | |
44, 13H, 1E5 | ||
32 | ||
33 | 1KS69, 54HSA | |
25Q8, 16L6W, B6CBQ | ||
4U5W5 | ||
36 | 16, 22O |
Extension to negative integers
Sum-product numbers can be extended to the negative integers by use of a signed-digit representation to represent each integer.
Programming example
The example below implements the sum-product function described in the definition above to search for sum-product numbers and cycles in Python.
def sum_product(x: int, b: int) -> int:
"""Sum-product number."""
sum_x = 0
product = 1
while x > 0:
if x % b > 0:
sum_x = sum_x + x % b
product = product * (x % b)
x = x // b
return sum_x * product
def sum_product_cycle(x: int, b: int) -> list[int]:
seen = []
while x not in seen:
seen.append(x)
x = sum_product(x, b)
cycle = []
while x not in cycle:
cycle.append(x)
x = sum_product(x, b)
return cycle
See also
- Arithmetic dynamics
- Dudeney number
- Factorion
- Happy number
- Kaprekar's constant
- Kaprekar number
- Meertens number
- Narcissistic number
- Perfect digit-to-digit invariant
- Perfect digital invariant
References
- ^ a b Proof that number of sum-product numbers in any base is finite, PlanetMath. Archived 2013-05-09 at the Wayback Machine by Raymond Puzio
- ^ Sloane, N. J. A. (ed.). "Sequence A038369 (Numbers n such that n = (product of digits of n) * (sum of digits of n).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- Arithmetic dynamics
- Base-dependent integer sequences