OFFSET
1,2
COMMENTS
Rotkiewicz defined these numbers and found the first 6 terms that are semiprimes (6, 14, 15, 35, 65, 119, and 377).
Křížek et al. named these numbers Rotkiewicz numbers, and proved that the following criteria are equivalent to the definition:
1) Numbers k such that c^sigma(k) == 1 (mod k) for all numbers c such that gcd(c, k) = 1.
2) Numbers k such that lambda(k) | sigma(k) where lambda is the Carmichael lambda function (A002322).
They also proved that:
1) If M(p) = 2^p-1 is a Mersenne prime (A000668) then 2^(p-2)*M(p) is a term.
2) If n is a term and, 2^k is the largest power of 2 that divides sigma(n), and F(m) = 2^(2^m) + 1 is a Fermat prime not dividing n such that m <= log_2(k+1) then n*F(m) is also a term.
REFERENCES
Michal Křížek, Florian Luca, and Lawrence Somer, 17 Lectures on Fermat Numbers: From Number Theory to Geometry, Springer-Verlag, New York, 2001, chapter 12, pp. 144-146.
Andrzej Rotkiewicz, Pseudoprime numbers and their generalizations, Student Association of Faculty of Sciences, University of Novi Sad, 1972.
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
Andrzej Rotkiewicz, Solved and unsolved problems on pseudoprime numbers, in: Applications of Fibonacci Numbers, Vol. 8 (ed. F. T. Howard), Kluwer Academic Publishers, Dordrecht, 1999, pp. 293-306.
MATHEMATICA
aQ[n_] := Divisible[DivisorSigma[1, n], CarmichaelLambda[n]]; Select[Range[560], aQ]
CROSSREFS
KEYWORD
nonn
AUTHOR
Amiram Eldar, Oct 25 2019
STATUS
approved