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The sequence in question is A296768 in the Online Encyclopedia. It starts with 1, 3, 5, 9, 11, 17, 24, 32, 36, 46, ... It is obtained by starting with the positive integers in order, (b(i)= i for all …

This question is a variant of the question posed by Brian Hopkins. Let $\operatorname{pdist}(n)$ be the number of partitions of $n$ into distinct parts. Also, let $\operatorname{pdist}(S, n)$ denote …

Here $n$ is a positive integer and $p(n)$ is the number of unrestricted partitions. Can one always find a subset $s$, of $\{1,2,\ldots,n\}$ such that the number of partitions of $n$ with parts fro …

Consider the two series defined by series 1: $$(1+x+x^2+x^3+...) (1+x^3+x^6+x^9...)(1+x^5+x^{10}+x^{15}+...)...$$ and series 2: $$1+x+x^2+x^5+x^7+...$$ Where the exponents 1,2,5,7,...are the pentagon …

Start with the product $$(1+x+x^2) (1+x^2)(1+x^3)(1+x^4)\cdots$$ (The first polynomial is a trinomial..The others are binomials..) Is it possible by changing some of the signs to get a series all of …

This question by Moshe Newman: How many different lattices are there on n points, that require exactly 3 elements to generate them? This sequence seems to start 0,0,1,0,4,3 (for n = 1 to 6) and seems …

For which $n$ is it possible to find a permutation of the numbers from $1$ to $n$ such that the sum of any two adjacent elements of the permutation is a prime? For example: For $n=4$ the permutation …

I'd like to have a Mathematica program to output planar partitions for small values. I seem to remember once writing such a program years ago using a program that appeared in an IEEE journal from the …

How can I prove the following? $$1-x+x^2+x^5-x^7-x^{12}+x^{15}-x^{22}-x^{26}+x^{35}-x^{40}+\dots \\= \prod_{i=1}^{\infty} [(1 - x^{8 i - 7}) (1 + x^{8 i - 6}) (1 + x^{8 i - 5}) (1 + x^{8 i - 4} …

This is my latest attempt to simplify an old problem of mine so much that the simplified problem can actually be answered. Starting with the generating function for unrestricted partitions: $$(1+x+x …

In his book of problems from elementary mathematics Hugo Steinhaus asked the following: does there exist for every positive integer N a sequence of real numbers $x_1,x_2,...,x_N$ such that for every …

This question is similar to another that I asked, but should be, I think, very much easier. Start with the generating function for unrestricted partitions and replace some of the plus signs with minu …

Let p(n) be the number of unrestricted partitions of n. p(0) is taken to be 1. Let set 1 and set 2 be two empty sets. Here's an algorithm. Put p(n) into set 1. On each successive step, k=1,2,3,..., …

Start with the product for unrestricted partitions: $(1+x+x^2+...)(1+x^2+x^4+...)(1+x^3+x^6+...)...$ Now replace some of the plus signs with minus signs and expand the product into a series. Is it …

Suppose $G$ is a $k$-generated finite group. Is there always a set of $k$ elements which generate the group and have a unimodal counting function? Background: The counting function, $f(n)$, is a fun …