# Equsum Partitions of Sets of Positive Integers +
## 2 Initial Intervals of \(\mathbb{Z}^{+}\)
As \(\mathcal{E}^{\prime}\) is a consesum partition of \([1,\ b-1]\) with defect \(\delta=1\), so \(\mathcal{C}^{\prime(m)}\) is a consesum \(b\)-partition of \([1,\,(2m-1)b-1]\) with defect \(\delta=1\). Hence  
**Theorem 3**. : _For any \(b\in\mathbb{Z}^{+},\ b\geq 2,\) and \(\delta\in\{0,\ 1\}\), the set \([1,\ n]\) has a consesum \(b\)-partition, with defect \(\delta\), for all \(n\in\mathbb{Z}^{+},\ n+\delta\equiv b\pmod{2b}\)._  
The next construction proceeds from equisum to consesum. **Lemma 2**. : _For any odd \(b\in\mathbb{Z}^{+},\ b\geq 3\), if \([1,\ n]\) has an equisum \(b\)-partition with defect \(\delta\), then \([1,\ n+b]\) and \([1,\ n+2b]\) have consesum \(b\)-partitions with defect \(\delta\)._  
**Proof. ** For any \(k\in\mathbb{Z}^{+},\ k\geq 2\), let \(b:=2k-1\). Let \(\mathcal{D}:=\{D_{i}|1\leq i\leq 2k-1=b\}\) be the \(b\)-partition of \([1,\ 2b]\) with blocks  
\[D_{i}:=\{i,\ 3k+i-2\},\ \ 1\leq i\leq k;\]  
\[D_{k+i}:=\{k+i,\ 2k+i-1\},\ 1\leq i\leq k-1.\]  
Clearly \(\Sigma D_{i}-3k+2\) runs through even members of \([1,\ 2k]\) as \(i\) runs through \([1,\ k]\), and \(\Sigma D_{k+i}-3k\) runs through odd members of \([1,\ 2k]\) as \(i\) runs through \([1,\ k-1]\). For each \(j\in\mathbb{Z}^{+},\ 3k\leq j\leq 5k-2\), there is an \(i\in\mathbb{Z}^{+},1\leq i\leq 2k-1=b\) such that \(D_{i}\) has block sum \(\Sigma D_{i}=j\). Thus there is a permutation \(\pi:[1,\ b]\to[1,\ b]\) such that  
\[\Sigma D_{\pi(i)}=3k+i-1,\ \ 1\leq i\leq b,\]  
so \(\mathcal{D}^{\pi}:=\left\{D_{\pi(i)}[1\leq i\leq b]\right\}\) is a uniform conoscsum \(b\)-partition of \([1,\ 2b]\). Let \(\mathcal{B}:=\{B_{i}|1\leq i\leq b\}\) be an equsum \(b\)-partition of \([1,\ n]\) with defect \(\delta\). Then, \(\mathcal{B}\oplus\mathcal{E}\) and \(\mathcal{B}\oplus\mathcal{D}^{\pi}\) are consesum \(b\)-partitions of \([1,\ n+b]\) and \([1,\ n+2b]\), respectively, both with defect \(\delta\). \(\Box\)  
The construction in the proof of Lemma 2, together with a modified version in which \(\mathcal{D}\) is replaced by \(\mathcal{D}-1\) throughout, yields  
**Lemma 3**. : _For any odd \(b\in\mathbb{Z}^{+},\ b\geq 3,\ and\ \delta\in\{0,\ 1\}\), the set \([1,\ n]\) has a consesum \(b\)-partition, with defect \(\delta\), for all \(\in\mathbb{Z}^{+},\ n+\delta\equiv 0\ (\mathrm{mod}\ b)\)._  
Forming direct sums with \(\mathcal{E}^{R}\) now yields  
**Theorem 4**. : _For any odd \(b\in\mathbb{Z}^{+},\ b\geq 3,\ and\ \delta\in\{0,\ 1\}\), the set \([1,\ n]\) has an equisum \(b\)-partition, with defect \(\delta\), for all \(n\in\mathbb{Z}^{+},\ n>b\), such that \(n+\delta\equiv 0\ (\mathrm{mod}\ b)\)._  
The interval \([1,\ n]\) can have an equisum \(b\)-partition only if at most one block is a singleton, so \(b\leq\lceil n/2\rceil\). Together, Theorems 2 and 4 establish  
**Theorem 5**. : _For any \(b,n\in\mathbb{Z}^{+},\ 2\leq b\leq\lceil n/2\rceil\), and \(\delta\in\{0,\ 1\}\), the set \([1,\ n]\) has an equisum partition of order \(b\), with defect \(\delta\), if (1) \(n+\delta\equiv 0\ (\mathrm{mod}\ 2b)\), or else if (2) \(n+\delta\equiv 0\ (\mathrm{mod}\ b)\), and \(b\) is odd._  
Let \(\mathbb{P}\) be the set of primes. As \(n\) and \(n+1\) are coprime, we have  
**Corollary 1**. : _For any \(a,n\in\mathbb{Z}^{+},\ p\in\mathbb{P}\), with \(n>p^{a}\), the set \([1,\ n]\) has an equisum partition of order \(p^{a}\), with defect \(\delta\in\{0,\ 1\}\), if and only if \(\ 2p^{a}\big{\lfloor}n(n+1)\,.\)_  
For any proper divisor \(b\) of \(n\) or \(n+1\), Theorem 5 settles when \([1,\ n]\) has an equisum partition of order \(b\). This leaves open the question of whether \([1,\ n]\) has an equisum partition of order \(b:=ac\) when \(a\geq 2,c\geq 2\) are coprime positive integers such that \(a|n,\ c|n+1\) and \(\ 2 ac\big{\lfloor}n(n+1)\). **Example 3**.