A329538
Odd composite numbers k such that A111076(k)^((k-1)/2) == -1 (mod k).
Original entry on oeis.org
29341, 1152271, 5481451, 14913991, 15247621, 36765901, 133800661, 178482151, 299736181, 579606301, 652969351, 702683101, 739444021, 743404663, 775368901, 3215031751, 4340265931, 5871134179, 8657319259, 9293756581, 12191597551, 13734086221, 14386156093, 19331388805
Offset: 1
-
f[1, lam_] = 1; f[n_, lam_] := If[n < 5, n - 1, Module[{k = 1}, While[GCD[k, n] > 1 || MultiplicativeOrder[k, n] < lam, k++]; k]]; aQ[n_] := CompositeQ[n] && Divisible[n - 1 , (lam = CarmichaelLambda[n])] && PowerMod[f[n, lam], (n - 1)/2, n] == n - 1; Select[Range[1, 6*10^6, 2], aQ] (* after the Charles R Greathouse IV at A111076 *)
A382791
Carmichael numbers with exactly 3 prime factors, p*q*r, such that p-1, q-1 and r-1 have an equal 2-adic valuation.
Original entry on oeis.org
8911, 29341, 314821, 410041, 1024651, 1152271, 5481451, 10267951, 14913991, 15247621, 36765901, 64377991, 67902031, 133800661, 139952671, 178482151, 188516329, 299736181, 362569201, 368113411, 395044651, 532758241, 579606301, 612816751, 620169409, 625482001, 652969351
Offset: 1
8911 = 7 * 19 * 67 is a term since it is a Carmichael number, it has exactly 3 prime factors, and 7 - 1 = 2*3, 19 - 1 = 2*3^2, and 67 - 1 = 2*3*11 all have 2-adic valuation 1.
- Amiram Eldar, Table of n, a(n) for n = 1..12071 (terms below 2^64)
- R. Balasubramanian and S. V. Nagaraj, The least witness of a composite number, International Workshop on Information Security, Springer, Berlin, Heidelberg, 1997, pp. 66-74.
- S. V. Nagaraj, Problems in Algorithmic Number theory, Ph.D. thesis, University of Madras, 1999. See Chapter 5, section 5.3, p. 43.
- Index entries for sequences related to Carmichael numbers.
-
q[n_] := Module[{f = FactorInteger[n]}, f[[;; , 2]] == {1, 1, 1} && SameQ @@ IntegerExponent[f[[;; , 1]] - 1, 2]];
Select[Cases[Import["https://oeis.org/A002997/b002997.txt", "Table"], {, }][[;; , 2]], q]
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isok(k) = if(!(k % 2) || isprime(k), 0, my(f = factor(k)); #f~ == 3 && k % lcm(znstar(k)[2]) == 1 && #Set(apply(x -> valuation(x-1, 2), f[,1])) == 1);
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