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Ruth-Aaron pairs containing Riesel or Sierpiński numbers. (English) Zbl 1448.11007

Given the prime-power decomposition of an integer \(n=p_1^{e_1}\cdots p_r^{e_r}\) we define \(S(n)=e_1p_1+\cdots +e_rp_r\) the sum (with multiplicities) of its factors. If \(S(n)=S(n+1)\), then \((n,n+1)\) forms a so-called Ruth-Aaron pair.
The authors use previous work by Nelson, Penney and Pomerance on these numbers, in which they provided a method to construct such pairs, in order to construct Ruth-Aaron pairs such that at least one of the members is a Riesel or a Sierpinski number.

MSC:

11A25 Arithmetic functions; related numbers; inversion formulas
11A51 Factorization; primality
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References:

[1] C. Nelson, D. E. Penney, and C. Pomerance, 714 and 715, J. Recreat. Math. 7 (1974), no. 2, 87-89.
[2] H. Riesel, N˚agra stora primtal, Elementa 39 (1956), 258-260.
[3] W. Sierpi´nski, Sur un probl‘eme concernant les nombres k2n+ 1, Elem. Math. 15 (1960), 73-74.
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