A095108 -id:A095108 - cp's OEIS frontend

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

19, 43, 67, 107, 127, 139, 163, 179, 211, 223, 227, 283, 307, 331, 347, 367, 379, 443, 463, 467, 487, 491, 499, 523, 547, 571, 587, 619, 631, 643, 683, 691, 727, 739, 787, 811, 823, 827, 859, 883, 907, 947, 967, 1019, 1051, 1087, 1123, 1163

Offset: 1

Antti Karttunen, Jun 01 2004

nonn

A. Karttunen and J. Moyer, C-program for computing the initial terms of this sequence

Intersection of A000040 and A095101. Complement of A095102 in A002145.

Cf. A095093, A095108 (diving indices).

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 *)

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

def A095103_list(n) :
    def is_Motzkin(n, k):
        s = 0
        for i in (1..k) :
            s += jacobi_symbol(i, n)
            if s < 0 : return false
        return true
    P = filter(is_prime, range(n+1)[3::4])
    return filter(lambda m: not is_Motzkin(m, m//2), P)
A095103_list(1163) # Peter Luschny, Aug 08 2012

a(n) = 4*A095273(n) + 3.

A095104 Diving index of the n-th 4k+3 prime (A002145(n)).

Original entry on oeis.org

0, 0, 0, 3, 0, 0, 3, 0, 0, 3, 0, 0, 0, 0, 7, 7, 0, 3, 0, 3, 0, 11, 0, 0, 3, 13, 61, 0, 0, 0, 0, 3, 3, 0, 3, 7, 0, 45, 3, 0, 0, 0, 0, 7, 7, 35, 0, 7, 35, 3, 0, 3, 3, 0, 3, 15, 0, 0, 3, 15, 3, 0, 0, 7, 3, 0, 45, 3, 0, 0, 3, 3, 7, 7, 0, 3, 0, 3, 0, 3, 0, 0, 7, 7, 0, 0, 0, 67, 0, 0, 3, 0

Offset: 1

Antti Karttunen, Jun 01 2004

nonn

Diving index of an odd number n is the first integer u > 1 where Sum_{i=1..u} J(i/n) results -1 and zero if never. Here J(i/n) is Jacobi symbol of i and n, which reduces to a Legendre symbol L(i/n) when n is a prime.

A. Karttunen and J. Moyer, C-program for computing the initial terms of this sequence

a(n)=A095105(n)+1 modulo A002145(n). Cf. A095106, A095108 (same sequence with zeros removed), A095269.

A095271 Diving index of A095101(n).

Original entry on oeis.org

3, 3, 10, 3, 3, 10, 7, 3, 7, 7, 3, 8, 8, 3, 11, 3, 34, 8, 19, 3, 10, 13, 61, 3, 7, 3, 7, 8, 3, 10, 3, 32, 7, 3, 58, 7, 3, 45, 3, 7, 3, 13, 31, 3, 8, 7, 3, 10, 7, 35, 3, 8, 7, 35, 3, 14, 8, 3, 13, 22, 3, 8, 3, 103, 7, 15, 3, 7, 40, 3, 7, 15, 55, 3, 113, 31, 3, 7, 13, 3, 118

Offset: 1

Antti Karttunen, Jun 01 2004

nonn

Antti Karttunen and J. Moyer, C-program for computing the initial terms of this sequence

Cf. A095101, A095108, A095269, A095275.

a(n) = A095269(A095275(n)). See comments at A095269.

cp's OEIS Frontend

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

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A095104 Diving index of the n-th 4k+3 prime (A002145(n)).

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A095271 Diving index of A095101(n).

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