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A275741 Sum of Wilson and Lerch remainders of n-th prime. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
2,2
COMMENTS
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
LINKS
John Blythe Dobson, A Characterization of Wilson-Lerch Primes, Integers, 16 (2016), A51.
René Gy, Generalized Lerch Primes, Integers 18 (2018), A10.
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.
FORMULA
a(n) = A002068(n) + A197631(n).
MATHEMATICA
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 *)
PROG
(PARI) 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)
CROSSREFS
Sequence in context: A068489 A088337 A195919 * A244846 A210415 A337275
KEYWORD
nonn
AUTHOR
Felix Fröhlich, Aug 07 2016
STATUS
approved

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Last modified May 27 11:47 EDT 2024. Contains 372858 sequences. (Running on oeis4.)