A362946 Positive integers that cannot be expressed as 1^e_1 + 2^e_2 + 3^e_3 ... + k^e_k with each exponent positive.

2, 4, 7, 11, 13, 19, 25, 31

Offset: 1

Author

Robert C. Lyons, Jul 05 2023

Keywords

• nonn
• hard

Comments

I conjecture that this list is finite.

Examples

			1 is not in the sequence because it's equal to 1^1.
3 is not in the sequence because it's equal to 1^1 + 2^1.
20 is not in the sequence because it's equal to 1^1 + 2^4 + 3^1.
29 is not in the sequence because it's equal to 1^1 + 2^2 + 3^1 + 4^2 + 5^1.
		

Crossrefs

Cf. A000217, A000330, A000537, A000538, A000539.

Programs

Python

from itertools import product
import math
max_term = 250
seq_set = set(range(1, max_term+1))
# Use the quadratic formula to calculate the maximum value for k,
# such that 1^1 + 2^1 + 3^1 + ... + k^1 is less than max_term.
max_k = int((-1 + math.sqrt(1 + 8 * max_term))/2.0) + 1
for k in range(1, max_k+1):
    list_of_exponent_ranges = [range(1,2)]
    for i in range(2, k+1):
        max_exponent = int(math.log(max_term, i))
        list_of_exponent_ranges.append(range(1, max_exponent+1))
    for exponents in product(*list_of_exponent_ranges):
        total = 0
        for i in range(1, k+1):
            total += int(i**exponents[i-1])
            if total > max_term:
                total = 0
                break
        if total in seq_set:
            seq_set.remove(total)
print(sorted(seq_set))