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A095103 4k+3 primes whose Legendre-vector is not valid Dyck-path.

%I A095103 #25 Jan 11 2022 12:42:17
%S A095103 19,43,67,107,127,139,163,179,211,223,227,283,307,331,347,367,379,443,
%T A095103 463,467,487,491,499,523,547,571,587,619,631,643,683,691,727,739,787,
%U A095103 811,823,827,859,883,907,947,967,1019,1051,1087,1123,1163
%N A095103 4k+3 primes whose Legendre-vector is not valid Dyck-path.
%H A095103 A. Karttunen and J. Moyer, <a href="/A095062/a095062.c.txt">C-program for computing the initial terms of this sequence</a>
%F A095103 a(n) = 4*A095273(n) + 3.
%t A095103 L = {}; Do[p = Prime[k]; If[Mod[p, 4] == 3 && Min[Table[Sum[JacobiSymbol[n, p], {n, 0, m}], {m, 0, p - 1}]] < 0, L = Append[L, p]], {k, 1, 192}]; L (* From Jonathan Sondow, Oct 25 2011 *)
%o A095103 (PARI) isok(m) = {my(s=0); if(m%4==3&&isprime(m), for(i=1, m-1, if((s+=kronecker(i, m))<0, return(1)))); 0; } \\ _Jinyuan Wang_, Jul 20 2020
%o A095103 (Sage)
%o A095103 def A095103_list(n) :
%o A095103     def is_Motzkin(n, k):
%o A095103         s = 0
%o A095103         for i in (1..k) :
%o A095103             s += jacobi_symbol(i, n)
%o A095103             if s < 0 : return false
%o A095103         return true
%o A095103     P = filter(is_prime, range(n+1)[3::4])
%o A095103     return filter(lambda m: not is_Motzkin(m, m//2), P)
%o A095103 A095103_list(1163) # _Peter Luschny_, Aug 08 2012
%Y A095103 Intersection of A000040 and A095101. Complement of A095102 in A002145.
%Y A095103 Cf. A095093, A095108 (diving indices).
%K A095103 nonn
%O A095103 1,1
%A A095103 _Antti Karttunen_, Jun 01 2004