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A324243
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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).
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1
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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
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OFFSET
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1,2
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COMMENTS
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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.
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REFERENCES
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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.
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LINKS
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MATHEMATICA
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aQ[n_] := Divisible[DivisorSigma[1, n], CarmichaelLambda[n]]; Select[Range[560], aQ]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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