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35 votes
8 answers
3k views

Examples of integer sequences coincidences

For the time being, the OEIS website contains almost $300000$ sequences. Each of these sequences is the mark of a specific mathematical concept. Sometimes two (or more) distinct concepts have the ...
33 votes
2 answers
856 views

A sequence potentially consisting of only integers

I will first ask the question which can be stated very simply. Afterwards I will explain some motivation and give references to related sequences. Consider the sequence defined by $$b_n = \frac{(...
John Machacek's user avatar
19 votes
1 answer
1k views

Is OEIS A007018 really a subsequence of squarefree numbers?

A comment in A007018 a(n) = a(n-1)^2 + a(n-1), a(0)=1 claims Subsequence of squarefree numbers (A005117). - Reinhard Zumkeller, Nov 15 2004 Is that really so? As far as I know, it is an open ...
joro's user avatar
  • 25.4k
18 votes
8 answers
2k views

Computationally challenging integer sequences

I wonder what are the examples of integer sequences, where only few elements are known and the researchers are still actively looking for the new terms. I think this discussion might be a good ...
Anton's user avatar
  • 1,625
12 votes
1 answer
634 views

Integrals of power towers

Let's assume $x\in[0,1]$, and restrict all functions of $x$ that we consider to this domain. Consider a sequence $\mathcal S_n$ of sets of functions, where $n^{\text{th}}$ element is the set of all ...
Vladimir Reshetnikov's user avatar
11 votes
1 answer
864 views

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 ...
joro's user avatar
  • 25.4k
7 votes
0 answers
210 views

My research paper involves computing additional terms of an existing OEIS sequence. Should I first amend the sequence or publish the results?

In the course of my research I computed terms of an existing OEIS sequence that are currently unknown. Having prepared my paper for publication, I am now faced with a (small) dilemma: Do I first ...
Klangen's user avatar
  • 1,962
5 votes
1 answer
345 views

Why does this "factorial sequence" appear in the OEIS?

For a reciprocal of a polynomial, $f = \frac{1}{p}$, we (presumably) may construct a sequence $(c_n)_{n=0}^\infty$ such that for all $N\ge 0$ $$f(k)k! = \sum_{n=0}^{N-1} c_n(k-n)! + O((k-N)!). $$ I ...
Zach Hunter's user avatar
  • 3,499
5 votes
0 answers
133 views

Formula and smallest solution for the A260711

Let $a(n)$ be A260711 without initial $0$ (i.e., numbers of the form $x^2 - y^2$ with $x > y$ where $x$ and $y$ are odd, $x + y$ is a power of $2$). The sequence begins with $$ 8, 16, 32, 48, 64, ...
Notamathematician's user avatar
2 votes
1 answer
131 views

Sequence that sums up to A224071

Let $a(n)$ be A224071 (i.e., number of Schroeder paths of semilength $n$ in which there are no $(2,0)$-steps at level $1$). Here $$ a(n) = \frac{1}{2(n+1)}\sum\limits_{k=0}^{n}(k+1)((-1)^{\left\...
Notamathematician's user avatar
2 votes
0 answers
163 views

Interesting conjecture by Sequence Machine

Let $a(n)$ be A344960 (i.e., position of binary complement of $n$-th word in A341258). By definition, in order to calculate $a(n)$, we need to know A341258. Below we will correspond this sequence with ...
Notamathematician's user avatar
2 votes
0 answers
72 views

Possible subsequence of the A110978

Let $a(n)$ be A110978 i.e. odd integers that are nonprime, such that there exist two factors of each number that when multiplied together in binary base, do not ever require the use of a "carry&...
Notamathematician's user avatar
2 votes
0 answers
327 views

Why can one compute the sum of divisors of $n$ without factoring $n$?

Question links to paper which states: $$ \sigma(n)= \frac{6}{n^2(n-1)}\sum_{k=1}^{n-1}(3n^2-10k^2)\sigma(k)\sigma(n-k) \qquad (1) $$ where $\sigma(n)$ is the sum of divisors of $n$. Another similar ...
joro's user avatar
  • 25.4k
1 vote
0 answers
100 views

Subsequence such that $c(a(n))=2^n$

Let $a(n)$ be A060831, i.e., $\sum\limits_{k=1}^{n}\operatorname{number of odd divisors of} k$. Let $$\ell(n)=\left\lfloor\log_2 n\right\rfloor$$ Let $$b(n,k)=2b(n,k-1)-2^{k-1}, b(n,0)=n$$ Let $c(n)$ ...
Notamathematician's user avatar
1 vote
0 answers
194 views

Closed form for partial sums of A103318

Let $a(n)$ be A103318, number of solutions $i$ in range $[0,n-1]$ to $i \equiv 0 \pmod {2^{n-i}}$: the sequence begins with $$1, 1, 2, 1, 2, 2, 2, 1, 2, 2, 3, 1, 2, 2, 2, 1, 2, 2, 3, 2$$ Also let's ...
Notamathematician's user avatar
0 votes
1 answer
104 views

Non-Wieferich primes with Euler quotient modulo $p$ two and alternating harmonic numbers

Let $b(n)$ denote the Euler quotient modulo $n$. In OEIS we have A128465 Numbers k such that k divides the numerator of alternating Harmonic number H'((k+1)/2) For $n>1$ we have $b(A128465(n))=2$. ...
joro's user avatar
  • 25.4k
0 votes
0 answers
190 views

On a A057985 without recursion

Let $a(n)$ be A057985 (i.e., start with $0$ and repeatedly substitute: $0\to01, 1\to12, 2\to0$). Let $\operatorname{wt}(n)$ be A000120 (i.e., number of ones in the binary expansion of $n$). Here $$ \...
Notamathematician's user avatar
0 votes
0 answers
107 views

Formula for individual term of the Proth numbers

Let $a(n)$ be A080075 i.e. Proth numbers: of the form $k2^m + 1$ for $k$ odd, $m \geqslant 1$ and $2^m > k$. The sequence begins with $$ 3, 5, 9, 13, 17, 25, 33, 41, 49, 57, 65, 81, 97, 113, 129 $$...
Notamathematician's user avatar