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A324243 Rotkiewicz numbers: numbers k such that a^sigma(k) == b^sigma(k) (mod k) for all pairs of numbers a, b such that gcd(a*b, k) = 1, where sigma(k) is the sum of divisors of k (A000203).

%I A324243 #62 Nov 02 2019 06:45:10
%S A324243 1,2,3,6,12,14,15,24,30,35,42,48,56,60,65,70,78,88,105,119,120,126,
%T A324243 130,140,168,182,190,195,210,224,238,240,248,255,260,264,270,280,312,
%U A324243 315,348,357,370,377,390,418,420,440,455,459,476,480,504,510,520,546,560
%N A324243 Rotkiewicz numbers: numbers k such that a^sigma(k) == b^sigma(k) (mod k) for all pairs of numbers a, b such that gcd(a*b, k) = 1, where sigma(k) is the sum of divisors of k (A000203).
%C A324243 Rotkiewicz defined these numbers and found the first 6 terms that are semiprimes (6, 14, 15, 35, 65, 119, and 377).
%C A324243 Křížek et al. named these numbers Rotkiewicz numbers, and proved that the following criteria are equivalent to the definition:
%C A324243 1) Numbers k such that c^sigma(k) == 1 (mod k) for all numbers c such that gcd(c, k) = 1.
%C A324243 2) Numbers k such that lambda(k) | sigma(k) where lambda is the Carmichael lambda function (A002322).
%C A324243 They also proved that:
%C A324243 1) If M(p) = 2^p-1 is a Mersenne prime (A000668) then 2^(p-2)*M(p) is a term.
%C A324243 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.
%D A324243 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.
%D A324243 Andrzej Rotkiewicz, Pseudoprime numbers and their generalizations, Student Association of Faculty of Sciences, University of Novi Sad, 1972.
%H A324243 Amiram Eldar, <a href="/A324243/b324243.txt">Table of n, a(n) for n = 1..10000</a>
%H A324243 Andrzej Rotkiewicz, <a href="https://doi.org/10.1007/978-94-011-4271-7_28">Solved and unsolved problems on pseudoprime numbers</a>, in: Applications of Fibonacci Numbers, Vol. 8 (ed. F. T. Howard), Kluwer Academic Publishers, Dordrecht, 1999, pp. 293-306.
%t A324243 aQ[n_] := Divisible[DivisorSigma[1, n], CarmichaelLambda[n]]; Select[Range[560], aQ]
%Y A324243 Cf. A000043, A000203, A000668, A002322, A019434, A236693.
%K A324243 nonn
%O A324243 1,2
%A A324243 _Amiram Eldar_, Oct 25 2019