A329238 - OEIS

%I #11 Dec 10 2023 17:44:41

%S 1,1,3,9,7,93,315,1,3855,13797,3,182361,41943,9709,9256395,34636833,

%T 31,117,1857283155,105,26817356775,102280151421,91,1497207322929,

%U 89756051247,1285,84973577874915,19065,4599,4885260612740877,18900352534538475,1,63,1101298153654301589

%N Carmichael quotients to base 2: a(n) = (2^lambda(2*n-1)-1)/(2*n-1), where lambda is the Carmichael lambda function (A002322).

%H Amiram Eldar, <a href="/A329238/b329238.txt">Table of n, a(n) for n = 1..1671</a>

%H Florian Luca, Min Sha, and Igor E. Shparlinski <a href="https://doi.org/10.4064/cm6910-3-2017">On two functions arising in the study of the Euler and Carmichael quotients</a>, Colloquium Mathematicum, Vol. 149, No. 2 (2017), pp. 179-192, <a href="https://arxiv.org/abs/1705.00388">arXiv preprint</a>, arXiv:1705.00388 [math.NT] (2017).

%H Min Sha, <a href="https://doi.org/10.1007/s10998-014-0079-3">The arithmetic of Carmichael quotients</a>, Periodica Mathematica Hungarica, Vol. 71, No. 1 (2015), pp. 11-23, <a href="https://doi.org/10.1007/s10998-017-0227-7">Correction to: The arithmetic of Carmichael quotients</a>, ibid., Vol. 76, No. 2 (2018), pp. 271-273, <a href="https://arxiv.org/abs/1108.2579">preprint</a>, arXiv:1108.2579v7 [math.NT] (2011-2017).

%H Chenhuang Wu, Zhixiong Chen, and Xiaoni Du, <a href="https://doi.org/10.1587/transfun.E95.A.1197">Binary threshold sequences derived from Carmichael quotients with even numbers modulus</a>, IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, Vol. 95, No. 7 (2012), pp. 1197-1199, <a href="https://www.researchgate.net/publication/258650079_Binary_Threshold_Sequences_Derived_from_Carmichael_Quotients_with_Even_Numbers_Modulus">alternative link</a>.

%e a(3) = (2^lambda(2*3 - 1) - 1)/(2*3 - 1) = (2^lambda(5) - 1)/5 = (2^4 - 1)/5 = 3.

%t a[n_] := (2^CarmichaelLambda[n] - 1)/n; Table[a[n], {n, 1, 67, 2}]

%Y Cf. A001226, A002322, A007663.

%K nonn

%O 1,3

%A _Amiram Eldar_, Nov 08 2019