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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. 6
 0, 13, 1356, 123229034, 79417031713, 97237045496594199, 166710337513971577670, 993090310179794898808058068, 60995221345838813484944512721637147449, 332049278209768881045237587717723153006704, 120846039713576242385812868532189241842793944235993733 (list; graph; refs; listen; history; text; internal format)
 OFFSET 2,2 COMMENTS 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 LINKS 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. FORMULA 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. EXAMPLE 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. MATHEMATICA 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 *) PROG (PARI) 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 CROSSREFS Cf. A007619, A197631, A197632. Sequence in context: A156641 A300592 A221690 * A222017 A238562 A196966 Adjacent sequences: A197627 A197628 A197629 * A197631 A197632 A197633 KEYWORD nonn AUTHOR Jonathan Sondow, Oct 16 2011 STATUS approved

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