A197631 -id:A197631 - cp's OEIS frontend

A002068 Wilson remainders: a(n) = ((p-1)!+1)/p mod p, where p = prime(n).

1, 1, 0, 5, 1, 0, 5, 2, 8, 18, 19, 7, 16, 13, 6, 34, 27, 56, 12, 69, 11, 73, 20, 70, 70, 72, 57, 1, 30, 95, 71, 119, 56, 67, 94, 86, 151, 108, 21, 106, 48, 72, 159, 35, 147, 118, 173, 180, 113, 131, 169, 107, 196, 214, 177, 73, 121, 170, 25, 277, 164, 231, 271, 259, 288, 110

Offset: 1

N. J. A. Sloane

If this is zero, p is a Wilson prime (see A007540).

Costa, Gerbicz, & Harvey give an efficient algorithm for computing terms of this sequence. - Charles R Greathouse IV, Nov 09 2012

R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001; see p. 29.
J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 244.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..2000
Edgar Costa, Robert Gerbicz, and David Harvey, A search for Wilson primes, arXiv:1209.3436 [math.NT], 2012.
C.-E. Froberg, Investigation of the Wilson remainders in the interval 3<=p<=50,000, Arkiv f. Matematik, 4 (1961), 479-481.
J. W. L. Glaisher, On the residues of the sums of products of the first p-1 numbers, and their powers, to modulus p^2 or p^3, Quart. J. Math. Oxford 31 (1900), 321-353.
K. Goldberg, A table of Wilson quotients and the third Wilson prime, J. London Math. Soc., 28 (1953), 252-256.
J. 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], 2011-2012.

Cf. A007540, A007619, A275741.

f:= p -> ((p-1)!+1 mod p^2)/p;
seq(f(ithprime(i)),i=1..1000); # Robert Israel, Jun 15 2014

Table[p=Prime[n]; Mod[((p-1)!+1)/p, p], {n,100}] (* T. D. Noe, Mar 21 2006 *)
Mod[((#-1)!+1)/#,#]&/@Prime[Range[70]] (* Harvey P. Dale, Feb 21 2020 *)

forprime(n=2, 10^2, m=(((n-1)!+1)/n)%n; print1(m, ", ")) \\ Felix Fröhlich, Jun 14 2014

a(n) = A007619(n) mod A000040(n).
a(n) + A197631(n) = A275741(n) for n > 1. - Jonathan Sondow, Jul 08 2019
a(n) = ( A027641(p-1)/A027642(p-1) + 1/p - 1 ) mod p, where p = prime(n), proved by Glashier (1900). - Max Alekseyev, Jun 20 2020

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

Original entry on oeis.org

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.

A197630 Lerch quotients of odd primes: ((Sum_{k=1..p-1} q_p(k)) - w_p)/p, where q_p(k) = (k^(p-1)-1)/p is a Fermat quotient, w_p = ((p-1)!+1)/p is a Wilson quotient, and p is the n-th prime, with n > 1.

Original entry on oeis.org

0, 13, 1356, 123229034, 79417031713, 97237045496594199, 166710337513971577670, 993090310179794898808058068, 60995221345838813484944512721637147449, 332049278209768881045237587717723153006704, 120846039713576242385812868532189241842793944235993733

Offset: 2

Jonathan Sondow, Oct 16 2011

nonn

Lerch proved that the Lerch quotient of any odd prime is an integer.
Is 13 the only Lerch quotient that is itself prime?
No other primes below 300,000 digits. - Charles R Greathouse IV, Nov 16 2011
Proof that a(n) is an integer for n >= 2: Note that ((p-1)!)^(p-1) = Product_{i=1..p-1} (1+i^(p-1)-1) == 1+Sum_{i=1..p-1} (i^(p-1)-1) (mod p^2). Write (p-1)! = kp-1, then ((p-1)!)^(p-1) == 1-(p-1)*kp == kp+1 == (p-1)!+2 (mod p^2). This gives Sum_{i=1..p-1} (i^(p-1)-1) == (p-1)!+1 (mod p^2), or Sum_{i=1..p-1} (i^(p-1)-1)/p == ((p-1)!+1)/p (mod p). - Jianing Song, Oct 15 2019

			a(3) = 13 because the 3rd prime is 5 and ((Sum_{k=1..4} q_5(k)) - w_5)/5 = (0 + 3 + 16 + 51 - 5)/5 = 13.

Michel Marcus, Table of n, a(n) for n = 2..75
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.
M. Lerch, Zur Theorie des Fermatschen Quotienten (a^(p-1)-1)/p = q(a), Math. Ann. 60 (1905), 471-490.
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 [math.NT], 2011-2012.
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.

Cf. A007619, A197631, A197632.

f[n_] := Block[{p = Prime[n]}, (Sum[(k^(p - 1) - 1)/p, {k, p - 1}] - ((p - 1)! + 1)/p)/p]; Array[f, 12, 2] (* Robert G. Wilson v, Dec 01 2016 *)

a(n)=my(p=prime(n),m=p-1); sum(k=1,m, k^m,-p-m!)/p^2 \\ Charles R Greathouse IV, Oct 18 2011

a(n) = ((Sum_{k=1..p-1} k^(p-1)) - p - (p-1)!)/p^2, where p is the n-th prime and n >= 2.

A275741 Sum of Wilson and Lerch remainders of n-th prime.

Original entry on oeis.org

1, 3, 10, 6, 6, 17, 15, 11, 25, 38, 9, 37, 47, 39, 86, 58, 107, 50, 101, 36, 98, 45, 123, 92, 170, 57, 80, 72, 158, 194, 194, 67, 78, 133, 120, 302, 144, 158, 128, 97, 91, 303, 76, 191, 139, 178, 302, 117, 242, 179, 335, 390, 362, 197, 290, 314, 327, 227, 429

Offset: 2

Felix Fröhlich, Aug 07 2016

nonn

a(n) = 0 if and only if prime(n) is in both A007540 and A197632, i.e., prime(n) is simultaneously a Wilson prime and a Lerch prime.
For n > 2, a(n) = 0 if and only if A027641(3*p-3) / A027642(3*p-3)-1 + 1/p == 0 (mod p^2), where p = prime(n) (cf. Dobson, 2016, theorem 2).
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

John Blythe Dobson, A Characterization of Wilson-Lerch Primes, Integers, 16 (2016), A51.
René Gy, Generalized Lerch Primes, Integers 18 (2018), A10.
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.

Cf. A002068, A007540, A197631, A197632.

a[n_] := Module[{p = Prime[n]}, Mod[((p-1)!+1)/p, p] + Mod[(Sum[(k^(p-1)-1)/p, {k, 1, p-1}] - ((p-1)!+1)/p)/p, p]];
Table[a[n], {n, 2, 60}] (* Jean-François Alcover, Feb 15 2019 *)

a002068(n) = my(p=prime(n)); ((p-1)!+1)/p % p
a197631(n) = my(p=prime(n), m=p-1); sum(k=1, m, k^m, -p-m!)/p^2 % p
a(n) = a002068(n) + a197631(n)

a(n) = A002068(n) + A197631(n).

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A002068 Wilson remainders: a(n) = ((p-1)!+1)/p mod p, where p = prime(n).

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

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A197630 Lerch quotients of odd primes: ((Sum_{k=1..p-1} q_p(k)) - w_p)/p, where q_p(k) = (k^(p-1)-1)/p is a Fermat quotient, w_p = ((p-1)!+1)/p is a Wilson quotient, and p is the n-th prime, with n > 1.

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A275741 Sum of Wilson and Lerch remainders of n-th prime.

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