A308963 Lerch pseudoprimes: composite numbers m such that Sum_{k=1..m-1} k^{m-1} - (m-1)! == m (mod m^2).

77, 161, 2261, 12839, 14231, 18668831, 1591100357

Offset: 1

Author

Amiram Eldar and Thomas Ordowski, Jul 03 2019

Keywords

• nonn
• more

Comments

According to Lerch's congruence (1905), if p is an odd prime, then Sum_{k=1..p-1} k^(p-1) - (p-1)! == p (mod p^2).

Equivalently, numbers m > 4 such that Sum_{k=1..m-1} k^(m-1) == m (mod m^2).

Equivalently, numbers m > 1 such that m*B_{m-1} == m (mod m^2), where B_k is the k-th Bernoulli number.

Equivalently, terms m of A121707 such that B_{m-1} == 1 (mod m).

Equivalently, numbers m > 1 such that A027641(m-1) == A027642(m-1) (mod m).

If m is a Lerch pseudoprime, then p-1 does not divide m-1 for every prime divisor p of m.

From M. F. Hasler, Jul 22 2019: (Start)

The Lerch primes A197632 satisfy Lerch's congruence "even" modulo p^3.

Up to a(7) all terms are either multiples of 7 or of 37, but not both. Will this pattern prevail?

We also note: a(1) = 7*11; a(2) = 7*(2*11 + 1) = a(1)/11*23; a(3) = 7*(2*7*23 + 1) = a(2)/23*17*19, a(5) = a(3)/17*107, i.e., a term in this subsequence has all but one of the prime factors of the preceding one. The subsequence (a(4), a(6), ...?) of terms divisible by 37 so far consists of semiprimes and therefore also has this property. (End)

Links

• Mathias Lerch, Zur Theorie des Fermatschen Quotienten (a^(p-1)-1)/p = q(a), Mathematische Annalen, Vol. 60, No. 4 (1905), pp. 471-490.
• Jonathan Sondow, Lerch quotients, Lerch primes, Fermat-Wilson quotients, and the Wieferich-non-Wilson primes 2, 3, 14771, In: Nathanson M. (eds) Combinatorial and Additive Number Theory. Springer Proceedings in Mathematics & Statistics, Vol. 101, Springer, New York, NY, 2014, pp. 243-255. Preprint: arXiv:1110.3113 [math.NT].

Crossrefs

A subsequence of A191677 and A121707.

Cf. A027641, A027642, A197630, A197632.

Programs

Mathematica

s={}; Do[If[CompositeQ[n] && Mod[Sum[PowerMod[k, n-1, n^2], {k, 1, n-1}] - (n-1)! - n, n^2] == 0, AppendTo[s, n]],{n,1,2500}] ; s

PARI

is_A308963(m)={sum(k=1,m-1,Mod(k,m^2)^(m-1))==m&&!isprime(m)&&m>4}
forcomposite(m=1,,is_A308963(m)&&print1(m",")) \\ Slow beyond 10000. - M. F. Hasler, Jul 22 2019

Extensions

a(6)-a(7) from Max Alekseyev, Jul 09 2019