A275741 - OEIS

%I #41 Nov 08 2024 08:00:26

%S 1,3,10,6,6,17,15,11,25,38,9,37,47,39,86,58,107,50,101,36,98,45,123,

%T 92,170,57,80,72,158,194,194,67,78,133,120,302,144,158,128,97,91,303,

%U 76,191,139,178,302,117,242,179,335,390,362,197,290,314,327,227,429

%N Sum of Wilson and Lerch remainders of n-th prime.

%C 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.

%C 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).

%C 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

%H John Blythe Dobson, <a href="http://www.integers-ejcnt.org/q51/q51.Abstract.html">A Characterization of Wilson-Lerch Primes</a>, Integers, 16 (2016), A51.

%H René Gy, <a href="http://math.colgate.edu/~integers/s10/s10.mail.html">Generalized Lerch Primes</a>, Integers 18 (2018), A10.

%H Jonathan Sondow, <a href="http://arxiv.org/abs/1110.3113">Lerch Quotients, Lerch Primes, Fermat-Wilson Quotients, and the Wieferich-non-Wilson Primes 2, 3, 14771</a>, in Proceedings of CANT 2011, arXiv:1110.3113 [math.NT], 2011-2012.

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

%t 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]];

%t Table[a[n], {n, 2, 60}] (* _Jean-François Alcover_, Feb 15 2019 *)

%o (PARI) a002068(n) = my(p=prime(n)); ((p-1)!+1)/p % p

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

%o a(n) = a002068(n) + a197631(n)

%Y Cf. A002068, A007540, A197631, A197632.

%K nonn

%O 2,2

%A _Felix Fröhlich_, Aug 07 2016