A197637 - OEIS

%I #17 May 21 2016 22:51:04

%S 0,1,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,6,6,6,6,7,7,7,7,7,7,8,8,9,9,9,9,

%T 9,9,10,10,10,10,11,11,12,12,12,12,13,13,13,13,13,13,14,14,14,14,14,

%U 14,15,15,16,16,16,16,16,16,17,17,17,17,18,18,19,19,19,19,19,19,20,20,20,20,21,21,21,21,21,21,22,22,22,22,22,22,22,22,23,23,23,23

%N Number of non-Wilson primes <= n.

%C The analog of pi(n) for non-Wilson primes.

%C An inverse function of A197636, as A197636(a(n)) = n if and only if n is a non-Wilson prime, i.e., a member of A197636.

%C Empirical evidence suggests that the sequence is unbounded, i.e., that A197636 is infinite, although no proof seems to be known. - _Felix Fröhlich_, May 18 2016

%H Felix Fröhlich, <a href="/A197637/b197637.txt">Table of n, a(n) for n = 1..10000</a>

%H E. Costa, R. Gerbicz and D. Harvey, <a href="http://dx.doi.org/10.1090/S0025-5718-2014-02800-7">A search for Wilson primes</a>, Mathematics of Computation, 83 (2014), 3071-3091 (arXiv:<a href="http://arxiv.org/abs/1209.3436">1209.3436</a> [math.NT], 2012).

%F a(A197636(n)) = n

%e There are 3 non-Wilson primes <= 8, namely 2, 3, and 7, so a(8) = 3.

%t nmax = 100; nonWilsonQ[p_] := Mod[((p-1)! + 1)/p, p] != 0; nonWilsonPrimes = Select[ Prime[ Range[nmax + 2]], nonWilsonQ]; a[n_] := Count[ nonWilsonPrimes, k_ /; k <= n]; Table[a[n], {n, 1, nmax}] (* _Jean-François Alcover_, Oct 10 2012 *)

%o (PARI) my(i=0); for(n=1, 50, if(ispseudoprime(n) && Mod((n-1)!, n^2)!=-1, i++); print1(i, ", ")) /* _Felix Fröhlich_, May 18 2016 */

%o (PARI) /* The following program is valid up to n = 2*10^13 (cf. Costa, Gerbicz, Harvey, 2014) */

%o my(w=[5, 13, 563], i=0); for(n=1, 200, for(k=1, #w, if(n==w[k], i++)); print1(primepi(n)-i, ", ")) /* _Felix Fröhlich_, May 18 2016 */

%Y Cf. A000720, A197636.

%K nonn

%O 1,3

%A _Jonathan Sondow_, Oct 19 2011