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1. Solutions of ϕ (n) = ϕ (n + k) and σ (n) = σ (n + k)

   https://doi.org/10.1093/imrn/rnaa218  

   Ford, Kevin (August 2020, International Mathematics Research Notices)

   null (Ed.)

   Abstract We show that for some even $$k\leqslant 3570$$ and all  $$k$$ with $442720643463713815200|k$, the equation $$\phi (n)=\phi (n+k)$$ has infinitely many solutions $$n$$, where $$\phi $$ is Euler’s totient function. We also show that for a positive proportion of all $$k$$, the equation $$\sigma (n)=\sigma (n+k)$$ has infinitely many solutions $$n$$. The proofs rely on recent progress on the prime $$k$$-tuples conjecture by Zhang, Maynard, Tao, and PolyMath. 

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2. Preparation and Characterization of N , N′ ‐Dialkyl‐1,3‐propanedialdiminium Chlorides, N , N′ ‐Dialkyl‐1,3‐propanedialdimines, and Lithium N , N′ ‐Dialkyl‐1,3‐propanedialdiminates

   https://doi.org/10.1002/hlca.202200152  

   Schmit, Claire E.; Girolami, Gregory S. (May 2023, Helvetica Chimica Acta)

   Abstract We describe convenient preparations ofN,N′‐dialkyl‐1,3‐propanedialdiminium chlorides,N,N′‐dialkyl‐1,3‐propanedialdimines, and lithiumN,N′‐dialkyl‐1,3‐propanedialdiminates in which the alkyl groups are methyl, ethyl, isopropyl, ortert‐butyl. For the dialdiminium salts, the N₂C₃backbone is always in thetrans‐s‐transconfiguration. Three isomers are present in solution except for thetert‐butyl compound, for which only two isomers are present; increasing the steric bulk of theN‐alkyl substituents shifts the equilibrium away from the (Z,Z) isomer in favor of the (E,Z), and (E,E) isomers. For the neutral dialdimines, crystal structures show that the methyl and isopropyl compounds adopt the (E,Z) form, whereas thetert‐butyl compound is in the (E,E) form. In aprotic solvents all four dialdimines (as well as the lithium dialdiminate salts) adoptcis‐s‐cisconformations in which there presumably is either an intramolecular hydrogen bond (or a lithium cation) between the two nitrogen atoms. 

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3. Smooth and analytic actions of SL(n,R) and SL(n,Z) on closed n-dimensional manifolds

   https://doi.org/10.1215/21562261-2024-0009  

   Fisher, David; Melnick, Karin (November 2024, Kyoto Journal of Mathematics)
4. Beyond N-Alkylation: Synthesis, Structure, and Function of N-Amino Peptides

   https://doi.org/10.1021/acs.accounts.4c00024  

   Angera, Isaac J; Wright, Madison M; Del_Valle, Juan R (May 2024, Accounts of Chemical Research)
5. Synthesis and crystallographic characterization of N-allyl-N-benzyl-4-methylbenzenesulfonamide

   https://doi.org/10.5155/eurjchem.11.3.245-249.2017  

   Stenfors, Brock Anton; Ngassa, Felix Nyuangem (September 2020, European Journal of Chemistry)

   null (Ed.)

   N-Benzyl-4-methylbenzenesulfonamides were prepared via a two-step synthetic process involving the treatment of 4-methylbenzenesulfonyl chloride with a primary amine to give the corresponding 4-methylbenzenesulfonamide. Benzylation of the sulfonamide affords the substituted N-benzyl-4-methylbenzenesulfonamides. The similarities between the two steps of synthesis lend credence to the development of a one-pot synthesis of substituted N-benzyl-4-methylbenzenesulfonamides from 4-methylbenzenesulfonyl chloride. This method was applied to the synthesis of N-allyl-N-benzyl-4-methylbenzenesulfonamide and characterized through spectroscopic and crystallographic means. The crystal structure of N-allyl-N-benzyl-4-methylbenzenesulfonamide was obtained by single-crystal X-ray diffraction. The crystal structure reveals an orthorhombic Pna21 space group with cell parameters a = 18.6919 (18) Å, b = 10.5612 (10) Å, c = 8.1065 (8) Å, V = 1600.3 (3) Å3 and Z = 4, T = 173.15 K, μ(MoKα) = 0.206 mm-1, Dcalc = 1.251 g/cm3, 14455 reflections measured (4.36° ≤ 2Θ ≤ 54.96°), 3619 unique (Rint = 0.0439, Rsigma = 0.0429) which were used in all calculations. The final R1 was 0.0428 (I > 2σ(I)) and wR2 was 0.1079 (all data). Molecules are linked through C-H···N hydrogen bonds and C-H···π interactions. 

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