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I'm curious to know if are in the literature analogous conjectures to the conjecture due to Mandl, I ask about these analogous conjectures for different sequences playing an important role in number theory, I evoke sequences closely related to prime numbers. You can see the conjecture due to Mandl for prime numbers from [1] or from the the first paragraph of [2] in page 1.

[EDIT: an inequality conjectured by Mandl asserts that

$$ \frac{np_n}{2} - \sum_{k \leq n} p_k \geq 0 $$ for every $n \geq 9$.]

Question. Do you know from the literature other similar/analogous conjectures to Mandl's conjecture for the comparison of the $n-th$ term of a number theoretic sequence $a_k$ and the summation $\sum_{1\leq k\leq n}a_k$? I am asking about arithmetic functions with a good mathematical content or playing an important role in theory and distribution of prime numbers, I evoke maybe Ramanujan primes, or maybe the sequence of square-free integers, or the sequence of powers of prime numbers,... I don't know. If you know it add the references for those works and I try to search in read these from the literature. In other case, can you illustrate how to get a conjecture similar to Mandl's conjecture for an interesting number theoretic sequence? Many thanks.

I think that Abel's identity should be an important tool to research candidates for conjectures. I would like to know how to do it in a professional way.

References:

[1] J. Barkley Rosser and L. Schoenfeld, Sharper Bounds for the Chebyshev Functions $\theta(x)$ and $\psi(x)$, Math. Of Computation, Vol. 29, Number 129 (January 1975).

[2] Christian Axler, On a Sequence Involving Prime Numbers, Journal of Integer Sequences, Vol 18 (2015), Article 15.7.6.

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  • $\begingroup$ I hope that my question is interesting. I hope that some user can provide references from the literature, or in case that my question isn't in the literature, the users can to illustrate how get a nice conjecture. If my post is welcome feel free to add/remove those tags more suitable. Other sequence with the best mathematical content seems the sequence of primorials, I think. $\endgroup$ – user142929 Aug 8 at 11:49
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    $\begingroup$ Question, or fishing expedition? In either case, I'd rather see a statement of the conjecture edited into the body of the question, than go chasing it offsite. $\endgroup$ – Gerry Myerson Aug 8 at 12:30
  • $\begingroup$ My intention is to learn, and share my ideas @GerryMyerson I', sorry many thanks $\endgroup$ – user142929 Aug 8 at 12:31
  • $\begingroup$ Hi all. Since my question is an exploration of inequalities involving $a_k$ and $\sum_{1\leq k\leq n}a_k$, where I evoke that $a_k$ is the general term of sequences (arithmetic functions) closely related to primes, maybe one can do a similar question than mine, now inspired in the so-called Selfridge's Conjecture. As reference that I know is the article Selfridge's Conjecture, from the encyclopedia Wolfram MathWorld. $\endgroup$ – user142929 Aug 9 at 9:55
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    $\begingroup$ I edited the conjecture statement into the body for you. Note that according to Axler, Mandl's conjecture has been proved. $\endgroup$ – Gerry Myerson Aug 9 at 10:07

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