Superincreasing sequence

In mathematics, a sequence of positive real numbers ${\displaystyle (s_{1},s_{2},...)}$ is called superincreasing if every element of the sequence is greater than the sum of all previous elements in the sequence.^[1][2]

Formally, this condition can be written as

${\displaystyle s_{n+1}>\sum _{j=1}^{n}s_{j}}$

for all n ≥ 1.

Program

The following Python source code tests a sequence of numbers to determine if it is superincreasing:

def is_superincreasing_sequence(sequence) -> bool:
    """Tests if a sequence is superincreasing."""
    total = 0
    result = True
    for n in sequence:
        print("Sum: ", total, "Element: ", n)
        if n <= total:
            result = False
            break
        total += n
    return result


sequence = [1, 3, 6, 13, 27, 52]
result = is_superincreasing_sequence(sequence)
print("Is it a superincreasing sequence? ", result)

This produces the following output:

Sum:  0 Element:  1
Sum:  1 Element:  3
Sum:  4 Element:  6
Sum:  10 Element:  13
Sum:  23 Element:  27
Sum:  50 Element:  52
Is it a superincreasing sequence?  True

Examples

• (1, 3, 6, 13, 27, 52) is a superincreasing sequence, but (1, 3, 4, 9, 15, 25) is not.^[2]
• The series a^x for a>=2

Properties

• Multiplying a superincreasing sequence by a positive real constant keeps it superincreasing.

See also

• Merkle–Hellman knapsack cryptosystem