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).

1, 2, 3, 6, 12, 14, 15, 24, 30, 35, 42, 48, 56, 60, 65, 70, 78, 88, 105, 119, 120, 126, 130, 140, 168, 182, 190, 195, 210, 224, 238, 240, 248, 255, 260, 264, 270, 280, 312, 315, 348, 357, 370, 377, 390, 418, 420, 440, 455, 459, 476, 480, 504, 510, 520, 546, 560

Offset: 1

Author

Amiram Eldar, Oct 25 2019

Keywords

• nonn

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.

Crossrefs

Cf. A000043, A000203, A000668, A002322, A019434, A236693.

Programs

Mathematica

aQ[n_] := Divisible[DivisorSigma[1, n], CarmichaelLambda[n]]; Select[Range[560], aQ]