A197630 -id:A197630 - cp's OEIS frontend

A197632 Lerch primes: odd primes that divide their Lerch quotients A197630.

3, 103, 839, 2237

Offset: 1

Jonathan Sondow, Oct 16 2011

Odd primes p such that Sum_{a=1..p-1} a^(p-1) - (p-1)! == p (mod p^3). (The congruence holds mod p^2 for any odd prime p; see Lerch (1905).)
Marek Wolf has computed that if a 5th Lerch prime p exists, then 4496113 < p < 18816869 or 18977773 < p < 32452867 or p > 32602373.
Can a number be simultaneously a Lerch prime and a Wilson prime A007540?
René Gy (see links) has shown that a number is simultaneously a Lerch prime and a Wilson prime if and only if it satisfies the congruence (p - 1)! + 1 == 0 (mod p^3). - John Blythe Dobson, Feb 23 2018
Named after the Czech mathematician Mathias Lerch (1860-1922). - Amiram Eldar, Jun 23 2021

			The 27th prime is 103, and A197631(27) = 0, so 103 is a member.

John Blythe Dobson, A note on Lerch primes, arXiv:1311.2242 [math.NT], 2014.
John Blythe Dobson, A Characterization of Wilson-Lerch Primes, Integers, Vol. 16 (2016), A51.
René Gy, Generalized Lerch Primes, Integers, Vol. 18 (2018), A10.
M. 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: M. Nathanson (ed.), 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], 2011-2012.

Cf. A000720, A007540, A197630, A197631, A308963.

Cases[Prime[Range[2, 500]], p_ /; Divisible[(Sum[(k^(p-1)-1)/p, {k, 1, p-1}] - ((p-1)! + 1)/p)/p, p]] (* Jean-François Alcover, Nov 21 2018 *)

is(p)=my(m=p-1,P=p^3); !sum(k=1, m, Mod(k,P)^m,-p-m!) && isprime(p) \\ Charles R Greathouse IV, Jun 18 2012

A197630(A000720(a(n))) == 0 (mod a(n)).

A197631(A000720(a(n))) = 0.

A197631 Lerch remainders: the Lerch quotient A197630 of the n-th prime p modulo p, where n > 1.

Original entry on oeis.org

0, 3, 5, 5, 6, 12, 13, 3, 7, 19, 2, 21, 34, 33, 52, 31, 51, 38, 32, 25, 25, 25, 53, 22, 98, 0, 79, 42, 63, 123, 75, 11, 11, 39, 34, 151, 36, 137, 22, 49, 19, 144, 41, 44, 21, 5, 122, 4, 111, 10, 228, 194, 148, 20, 217, 193, 157, 202, 152, 87, 93, 30, 219

Offset: 2

Jonathan Sondow, Oct 16 2011

			a(3) = A197630(3) mod Prime(3) = 13 mod 5 = 3.

J. B. Dobson A note on Lerch primes, arXiv:1311.2242 [math.NT], 2014.
J. B. Dobson A Characterization of Wilson-Lerch Primes, Integers, 16 (2016), A51.
Jonathan Sondow, Lerch Quotients, Lerch Primes, Fermat-Wilson Quotients, and the Wieferich-non-Wilson Primes 2, 3, 14771, in Proceedings of CANT 2011, arXiv:1110.3113 [math.NT], 2011-2012.
Jonathan Sondow, Lerch Quotients, Lerch Primes, Fermat-Wilson Quotients, and the Wieferich-non-Wilson Primes 2, 3, 14771, Combinatorial and Additive Number Theory, CANT 2011 and 2012, Springer Proc. in Math. & Stat., vol. 101 (2014), pp. 243-255.

Cf. A197630, A197632.

a(n) = my(p=prime(n), m=p-1); sum(k=1, m, k^m, -p-m!)/p^2 % p;
vector(100, n, a(n+1)) \\ Altug Alkan, Nov 22 2015

a(n) = A197630(n) mod Prime(n), with n >= 2.

A222207 Morley quotients: (2^(2p-2) - (-1)^((p-1)/2)binomial(p-1,(p-1)/2)) / p^3, where p = prime(n) and n >= 3.

Original entry on oeis.org

2, 12, 788, 7636, 874202, 10018884, 1445893544, 2954512034024, 38700329118256, 93229749133527532, 17540746936557672236, 243284404062970619608, 47694250379410432495952, 136236017676683906365850456, 404504597532158799519693872144, 5856120097210409121404621878992, 18102352585707069737371994385420772, 3894254646848417473467131712404310728

Offset: 3

Jonathan Sondow, Feb 22 2013

nonn

Morley (1894/95) proved 2^(2*p-2) == (-1)^((p-1)/2)*binomial(p-1,(p-1)/2) mod p^3 for all primes p > 3.

Morley quotients are even, since 2^(2*p-2) and binomial(p-1,(p-1)/2) are even and p^3 is odd.

			prime(3) = 5, so a(3) = (2^(2*5-2) - (-1)^((5-1)/2)*binomial(5-1,(5-1)/2))/5^3 = (2^8 - binomial(4,2))/5^3 = (256-6)/125 = 2.

Vincenzo Librandi, Table of n, a(n) for n = 3..200
C. Aebi, G. Cairns, Morley’s other miracle, Math. Mag., 85 (2012), 205-211.
F. Morley, Note on the Congruence 2^4n == (-1)^n*(2n)!/(n!)^2 where 2n+1 is a prime, Annals of Mathematics, Vol. 9 (1894 - 1895), pp. 168-170.

Cf. A007619, A007663, A034602, A197630, A197633.

m[p_] := (2^(2*p-2) - (-1)^((p-1)/2)*Binomial[p-1, (p-1)/2])/p^3; Table[ m[ Prime[n]], {n, 3, 20}]

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

Original entry on oeis.org

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

Offset: 1

Amiram Eldar and Thomas Ordowski, Jul 03 2019

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)

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].

A subsequence of A191677 and A121707.

Cf. A027641, A027642, A197630, A197632.

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

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

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

A229833 1/p^3 * numerator((sum_{j=1..p-1} j^(p-1)) - p*Bernoulli(p-1)) with p = prime(n).

Original entry on oeis.org

17, 1175, 67232195, 1282936297603, 171594913930219489, 368517627392700495869, 259067037992493907740808871, 63098504840897942292160460526547792021, 4948605372033572359620687688871811178548595, 169413083241708480729625174442441002390094469490644564301, 90165569601996395473034926239938857618854516797194687641929891

Offset: 3

Jonathan Sondow, Oct 16 2013

nonn

Sum_{j=1..p-1} j^(p-1) == p*Bernoulli(p-1) (mod p^3) for prime p > 3 (see formulas (8) and (10) in "Lerch Quotients, ..."), so a(n) is an integer for n > 2.

			Prime(3) = 5 and 1/5^3 * numerator((sum_{j=1..4} j^4) - 5*Bernoulli(4)) = 1/125 * numerator(354 - 5*(-1/6)) = 2125/125 = 17, so a(3) = 17.

J. Sondow, Lerch Quotients, Lerch Primes, Fermat-Wilson Quotients, and the Wieferich-non-Wilson Primes 2, 3, 14771, in Proceedings of CANT 2011, arXiv:1110.3113
J. Sondow, Lerch Quotients, Lerch Primes, Fermat-Wilson Quotients, and the Wieferich-non-Wilson Primes 2, 3, 14771, Combinatorial and Additive Number Theory, CANT 2011 and 2012, Springer Proc. in Math. & Stat., vol. 101 (2014), pp. 243-255.
Index entries for sequences related to Bernoulli numbers.

Cf. A197630.

Table[p = Prime[n]; Numerator[ Sum[j^(p - 1), {j, 1, p - 1}] - p*BernoulliB[p - 1]]/p^3, {n, 3, 13}]

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A197632 Lerch primes: odd primes that divide their Lerch quotients A197630.

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A197631 Lerch remainders: the Lerch quotient A197630 of the n-th prime p modulo p, where n > 1.

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A222207 Morley quotients: (2^(2*p-2) - (-1)^((p-1)/2)*binomial(p-1,(p-1)/2)) / p^3, where p = prime(n) and n >= 3.

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A308963 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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A229833 1/p^3 * numerator((sum_{j=1..p-1} j^(p-1)) - p*Bernoulli(p-1)) with p = prime(n).

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A222207 Morley quotients: (2^(2p-2) - (-1)^((p-1)/2)binomial(p-1,(p-1)/2)) / p^3, where p = prime(n) and n >= 3.