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A308963
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Lerch pseudoprimes: composite numbers m such that Sum_{k=1..m-1} k^{m-1} - (m-1)! == m (mod m^2).
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1
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OFFSET
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1,1
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COMMENTS
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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.
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)
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LINKS
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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].
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MATHEMATICA
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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
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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