# Find minimum integer such that any integer \in [1, n] can be constructed from its consequent subsums

For example, here's (SPOILERS) breakdown for $$1143$$, which is the solution for $$n = 9$$

1. $$\underline{1}143$$
2. $$\underline{11}43$$
3. $$114\underline{3}$$
4. $$11\underline{4}3$$
5. $$1\underline{14}3$$
6. $$\underline{114}3$$
7. $$11\underline{43}$$
8. $$1\underline{143}$$
9. $$\underline{1143}$$

The questions are:

• are there more anomalies like $$18$$, what are they?
• are most solutions like that: $$111 \dots some\ digits\ within \pm1$$?

Those digits at the tail seem to grow and they cannot be more than $$9$$

### Appendix A: Brute-force solution in JavaScript

function partialSums(numbers) {
let sum = 0;
const result = [0];
for (const number of numbers) {
result.push(sum += number);
}
return result;
}

function solve(maxSum) {
for (let n = 1; true; n++) {
const s = n.toString();
// simple shortcut for speed up
if (s.includes('0')) continue;
const digits = s.split('').map(Number);
const sums = new Set();
const partials = partialSums(digits);
for (let start = 0; start < digits.length; start++) {
for (let end = start + 1; end <= digits.length; end++) {
const subSum = partials[end] - partials[start];
// sum will only increase further in the inner loop
// so we can just break from it
if (subSum > maxSum) break;
if (sums.size === maxSum) return n;
}
}
}
}


### Appendix B: Brute-force results (More SPOILERS)

$$\begin {array} {r|r} \ n & solution\\ \hline 1 & 1 \\ 2 & 11 \\ 3 & 12 \\ 4 & 112 \\ 5 & 113 \\ 6 & 132 \\ 7 & 1114 \\ 8 & 1133 \\ 9 & 1143 \\ 10 & 11134 \\ 11 & 11144 \\ 12 & 11154 \\ 13 & 11443 \\ 14 & 111155 \\ 15 & 111165 \\ 16 & 111544 \\ 17 & 111554 \\ 18 & 256318 \\ 19 & 1111555 \\ 20 & 1111655 \\ 21 & 1111665 \\ 22 & 1115554 \\ 23 & 1194332 \\ 24 & 11111766 \\ 25 & 11111776 \\ 26 & 11116565 \\ 27 & 11116665 \\ 28 & 12337741 \\ 29 & 12377441 \\ 30 & 111116766 \\ \end {array}$$

• oeis.org/A296447 Commented Jan 28, 2020 at 15:11
• and OEIS answers one question: $a(59) = 213388888552$ Commented Jan 28, 2020 at 15:38
• Shame on me, I should have tried to search in OEIS Commented Jan 28, 2020 at 18:21
• @almagest BTW there's a mistake on their site. They say "the latter being the second term of the sequence not to start with 1; a(59) = 213388888552" and reference a table with a(51) also starting not from 1. Actually, it's exactly a(59) with one 8 removed Commented Jan 28, 2020 at 18:46
• @kirilloid You should let them know. See here Commented Jan 28, 2020 at 18:50