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8 questions
6
votes
0
answers
171
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An inequality involving integer partitions
For integers $n\ge k\ge0$, let $p(n,k)$ denote the number of ways to write $n$ as a sum of $k$ positive integers (repetition allowed). For example, $p(6,3)=3$ since
$$6=1+1+4=1+2+3=2+2+2.$$
QUESTION. ...
- 15.6k
3
votes
0
answers
120
views
Sequence which is related to the binary expansion of $n$ and partition numbers
Let $p(n)$ be A000041 i.e. the number of partitions of $n$ (the partition numbers).
Let
$$
\ell(n)=\left\lfloor\log_2 n\right\rfloor
$$
Let $\operatorname{wt}(n)$ be A000120 i.e. number of $1$'s in ...
- 4,948
1
vote
0
answers
100
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Conjecture on numbers $k$ having only one partition into parts with same binary weight as a binary weight of $k$
Let $\operatorname{tr}(n)$ be A007814, number of trailing zeros in the binary representation of $n$.
Also, let $\operatorname{ntr}(n)$ be A086784, number of non-trailing zeros in the binary ...
- 4,948
3
votes
1
answer
92
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Partition of $(2^{n+1}+1)2^{2^{n-1}+n-1}-1$ into parts with binary weight equals $2^{n-1}+n$
Let $\operatorname{wt}(n)$ be A000120, i.e., number of $1$'s in binary expansion of $n$ (or the binary weight of $n$).
Let $a(n,m)$ be the sequence of numbers $k$ such that $\operatorname{wt}(k)=m$. ...
- 34.4k
2
votes
0
answers
137
views
Writing integers as sequences of products by 2 and integer divisions by 3
For any integer, we consider its decompositions into sequences of products by $2$ and integer division by $3$.
For instance:
$$
100 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \...
- 1,444
11
votes
1
answer
864
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Up to $10^6$: $\sigma(8n+1) \mod 4 = OEIS A001935(n) \mod 4$ (Number of partitions with no even part repeated )
Up to $10^6$:
$\sigma(8n+1) \mod 4 = OEIS A001935(n) \mod 4$
A001935 Number of partitions with no even part repeated
Is this true in general?
It would mean relation between restricted partitions ...
- 19.1k
2
votes
0
answers
154
views
Equi-distribution of the parity of partitions
The integer partition function $p(n)$ has a generating function given by
$$\frac1{(q)_{\infty}}=\sum_{n=0}^{\infty}p(n)q^n$$
with $(q)_{\infty}=\prod_{m=1}^{\infty}(1-q^m)$. The long-standing problem ...
- 43.2k
13
votes
0
answers
718
views
Is "OEIS A001935 Number of partitions with no even part repeated" efficiently computable $\mod 4$?
Is A001935 Number of partitions with no even part repeated efficiently computable $\mod 4$?
I am interested because of this relation with sum of divisors of $8n+1$.
$\sigma(8n+1) \equiv A001935(n) \...
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