A001226 -id:A001226 - cp's OEIS frontend

A329238 Carmichael quotients to base 2: a(n) = (2^lambda(2n-1)-1)/(2n-1), where lambda is the Carmichael lambda function (A002322).

1, 1, 3, 9, 7, 93, 315, 1, 3855, 13797, 3, 182361, 41943, 9709, 9256395, 34636833, 31, 117, 1857283155, 105, 26817356775, 102280151421, 91, 1497207322929, 89756051247, 1285, 84973577874915, 19065, 4599, 4885260612740877, 18900352534538475, 1, 63, 1101298153654301589

Offset: 1

Amiram Eldar, Nov 08 2019

nonn

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

Amiram Eldar, Table of n, a(n) for n = 1..1671
Florian Luca, Min Sha, and Igor E. Shparlinski On two functions arising in the study of the Euler and Carmichael quotients, Colloquium Mathematicum, Vol. 149, No. 2 (2017), pp. 179-192, arXiv preprint, arXiv:1705.00388 [math.NT] (2017).
Min Sha, The arithmetic of Carmichael quotients, Periodica Mathematica Hungarica, Vol. 71, No. 1 (2015), pp. 11-23, Correction to: The arithmetic of Carmichael quotients, ibid., Vol. 76, No. 2 (2018), pp. 271-273, preprint, arXiv:1108.2579v7 [math.NT] (2011-2017).
Chenhuang Wu, Zhixiong Chen, and Xiaoni Du, Binary threshold sequences derived from Carmichael quotients with even numbers modulus, IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, Vol. 95, No. 7 (2012), pp. 1197-1199, alternative link.

Cf. A001226, A002322, A007663.

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

A329706 Odd numbers k such that Sum_{j=1..(k-1)/2, gcd(j,k)=1} 1/j == -2q_2(k) + kq_2(k)^2 (mod k^3), where q_2(k) = (2^phi(k) - 1)/k is the Euler quotient of k to base 2.

Original entry on oeis.org

1, 3, 597, 609, 1791, 2035, 3403, 3701, 4263, 27515, 27955

Offset: 1

Amiram Eldar, Feb 28 2020

Emma Lehmer proved that Sum_{j=1..(p-1)/2} 1/j == -2*q_2(p) + p*q_2(p)^2 (mod p^2) for all odd primes p.
Tianxin Cai generalized Lehmer's congruence and proved that Sum_{j=1..(k-1)/2, gcd(j,k)=1} 1/j == -2*q_2(k) + k*q_2(k)^2 (mod k^2) for all odd numbers k.
This sequence includes the odd numbers k for which the congruence is still valid when (mod k^2) is being replaced with (mod k^3).
The prime terms are 3, 3701, ...
No more terms below 147000.

Tianxin Cai, A congruence involving the quotients of Euler and its applications (I), Acta Arithmetica, Vol. 103, No. 4 (2002), pp. 313-320.
Tianxin Cai, A Generalization of E. Lehmer's Congruence and Its Applications, in: ChaohuaJia and Kohji Matsumoto (eds.), Analytic Number Theory, Springer, Boston, MA, 2002, pp. 93-98.
Emma Lehmer, On congruences involving Bernoulli numbers and the quotients of Fermat and Wilson, Annals of Mathematics, Second Series, Vol. 39, No. 2 (1938), pp. 350-360, alternative link.

Cf. A000010, A001226.

q[n_] := (2^EulerPhi[n] - 1)/n; Select[Range[1, 2100, 2], Divisible[Numerator[Sum[Boole @ CoprimeQ[j, #]/j, {j, 1, (# - 1)/2}] + 2*q[#] - #*q[#]^2], #^3] &]

cp's OEIS Frontend

A329238 Carmichael quotients to base 2: a(n) = (2^lambda(2n-1)-1)/(2n-1), where lambda is the Carmichael lambda function (A002322).

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A329706 Odd numbers k such that Sum_{j=1..(k-1)/2, gcd(j,k)=1} 1/j == -2q_2(k) + kq_2(k)^2 (mod k^3), where q_2(k) = (2^phi(k) - 1)/k is the Euler quotient of k to base 2.

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cp's OEIS Frontend

A329238 Carmichael quotients to base 2: a(n) = (2^lambda(2*n-1)-1)/(2*n-1), where lambda is the Carmichael lambda function (A002322).

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A329706 Odd numbers k such that Sum_{j=1..(k-1)/2, gcd(j,k)=1} 1/j == -2*q_2(k) + k*q_2(k)^2 (mod k^3), where q_2(k) = (2^phi(k) - 1)/k is the Euler quotient of k to base 2.

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A329238 Carmichael quotients to base 2: a(n) = (2^lambda(2n-1)-1)/(2n-1), where lambda is the Carmichael lambda function (A002322).

A329706 Odd numbers k such that Sum_{j=1..(k-1)/2, gcd(j,k)=1} 1/j == -2q_2(k) + kq_2(k)^2 (mod k^3), where q_2(k) = (2^phi(k) - 1)/k is the Euler quotient of k to base 2.