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.

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

Offset: 1

Author

Amiram Eldar, Feb 28 2020

Keywords

• nonn
• more

Comments

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.

Links

• 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.

Crossrefs

Cf. A000010, A001226.

Programs

Mathematica

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] &]