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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)
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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