Timeline for Only finitely many values of the symmetric functions of $1/1,1/2,\ldots,1/n$ are $2$-adic integers (?)
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| Sep 21, 2017 at 19:11 | comment | added | Noam D. Elkies | I might not get to check this soon, but I gave gp code which runs in well under a minute (and gp is free software) so anybody who needs these numbers can just reproduce my calculation. | |
| Sep 21, 2017 at 11:49 | comment | added | Todd Trimble | @NoamD.Elkies I had to modify your first comment because the original was causing it and other comments to extend way off to the right. Hopefully I didn't miscount, and apologies if I did. | |
| Sep 20, 2017 at 21:36 | history | edited | François G. Dorais |
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| Aug 31, 2017 at 12:11 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
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| S Mar 10, 2017 at 17:13 | history | bounty ended | CommunityBot | ||
| S Mar 10, 2017 at 17:13 | history | notice removed | CommunityBot | ||
| Mar 8, 2017 at 2:30 | comment | added | Noam D. Elkies | The next 255 values are all $8$ except for $n=439$, again with the minimum at $k=2$ (this time with valuation $-5$, not $-3$). | |
| Mar 8, 2017 at 2:26 | comment | added | Noam D. Elkies | Experimentally this seems plausible. Here are the minima $\min_{k>0} (-v_2(H(n,k))$ for $n=1,2,3,\ldots,256$: $010221132333133 (4^{11}) 34444(5^{22})4(5^9) (6^{45})3(6^{18})(7^{91})3(7^{36})8$ where $(n^k)$ indicates a string of $k$ consecutive $n$'s. The stray 3's for $n=109$ and $n=219$ both occur at $k=2$. gp code: f(n,p)=p=prod(i=1,n,1+x/i); vector(n,j,-valuation(polcoeff(p,j),2)); vector(256,n,f(n)) | |
| Mar 4, 2017 at 9:00 | answer | added | js21 | timeline score: 1 | |
| Mar 2, 2017 at 20:00 | comment | added | Greg Martin | Hmmm, yes you're right, sorry. I don't know if anything can be gotten out of modifying my comment.... | |
| Mar 2, 2017 at 18:14 | comment | added | user40023 | @GregMartin For $k = 2$ and $n = 6$ both the terms $1/(2 \cdot 4)$ and $1/(4 \cdot 6)$ have minimal $2$-adic valuations. | |
| Mar 2, 2017 at 17:47 | comment | added | Greg Martin | Note that if $1\le k\le n/2$, then there is a unique term in the summation with the maximal power of $2$ in the denominator, and hence the sum is not a $2$-adic integer in those cases. (This is a generalization of the proof for $H(n,1)$.) | |
| S Mar 2, 2017 at 15:45 | history | bounty started | CommunityBot | ||
| S Mar 2, 2017 at 15:45 | history | notice added | user40023 | Authoritative reference needed | |
| Feb 27, 2017 at 9:36 | history | asked | user40023 | CC BY-SA 3.0 |