A095093 -id:A095093 - cp's OEIS frontend

A095355 Ratio A095106(n)/A095093(n) rounded down.

0, 0, 0, 3, 3, 5, 15, 13, 10, 13, 25, 22, 97, 85, 203, 359, 625, 1067, 1880, 3166, 6068

Offset: 1

Antti Karttunen, Jun 12 2004

nonn

This is the average diving index for those 4k+3 primes in range ]2^n,2^(n+1)] that "dive". See A095103.
The ratios before rounding are: 0, 0, 0, 3, 3, 5.666667, 15.666667, 13.166667, 10.142857, 13.926829, 25.805195, 22.118881, 97.536585, 85.237736, 203.39802, 359.470768, 625.039342, 1067.145123, 1880.907721, 3166.124599, 6068.683879.
Ratio (A095106(n)/A095093(n))/(A095109(n)/A095091(n)) starts as follows: 0, 0, 0, 1, 0.5, 0.428571, 0.4, 0.387097, 0.308824, 0.277027, 0.248387, 0.215361, 0.213383, 0.191474, 0.178036, 0.169496, 0.156814, 0.148329, 0.141456, 0.134383.

A095356 gives the same ratios rounded to nearest integer. A095359 gives similar ratios computed for all 4k+3 integers.

a(n) = 0 if A095093(n) is 0, otherwise a(n) = floor(A095106(n)/A095093(n)).

A095356 Ratio A095106(n)/A095093(n) rounded to nearest integer.

Original entry on oeis.org

0, 0, 0, 3, 3, 6, 16, 13, 10, 14, 26, 22, 98, 85, 203, 359, 625, 1067, 1881, 3166, 6069

Offset: 1

Antti Karttunen, Jun 12 2004

Cf. A095355, where the same ratios are given round down.

a(n) = 0 if A095093(n) is 0, otherwise a(n) = round(A095106(n)/A095093(n)).

A095008 Number of 4k+3 primes (A002145) in range ]2^n,2^(n+1)].

Original entry on oeis.org

1, 1, 1, 3, 3, 7, 13, 22, 37, 71, 128, 231, 440, 807, 1519, 2872, 5371, 10204, 19341, 36759, 70179, 134241, 256856, 492936, 947272, 1822615, 3513691, 6781495, 13103816, 25348667, 49092241, 95168205, 184661253, 358636497, 697094872, 1356052491, 2639893495, 5142817901

Offset: 1

Antti Karttunen and Labos Elemer, Jun 01 2004

Antti Karttunen and John Moyer, C-program for computing the initial terms of this sequence.
Index entries for sequences related to occurrences of various subsets of primes in range ]2^n,2^(n+1)].

Cf. A002145, A036378, A095007, A095010, A095012, A095092, A095093.

a(n) = A036378(n) - A095007(n) = A095010(n) + A095012(n) = A095092(n) + A095093(n).

a(34)-a(38) from Amiram Eldar, Jun 12 2024

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

Original entry on oeis.org

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.

A095091 Number of 4k+3 integers in range [2^n, 2^(n+1)] whose Jacobi-vector is not a valid Motzkin-path (A095101).

Original entry on oeis.org

0, 0, 0, 1, 2, 7, 15, 31, 68, 148, 310, 664, 1345, 2768, 5673, 11605, 23665, 48035, 97599, 197383

Offset: 1

Antti Karttunen, Jun 01 2004

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

Cf. A095093.

a(n) = 2^(n-2) - A095090(n) for n > 1.

A095092 Number of 4k+3 primes whose Legendre-vector is a Dyck-path (A095102) in range ]2^n,2^(n+1)].

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 7, 10, 16, 30, 51, 88, 153, 277, 509, 905, 1660, 3079, 5535, 10234, 19053

Offset: 1

Antti Karttunen, Jun 01 2004

A. Karttunen and J. Moyer, C-program for computing the initial terms of this sequence
Index entries for sequences related to occurrences of various subsets of primes in range ]2^n,2^(n+1)]

Cf. A095090, A095102.

is(m) = {if(!isprime(m), return(0)); my(s=0); for(i=1, m-1, if((s+=kronecker(i, m))<0, return(0))); 1; }
a(n) = {my(c=0); forstep(m=2^n+3*(n>1), 2^(n+1), 4, c+=is(m)); c; } \\ Jinyuan Wang, Jul 20 2020

a(n) = A095008(n) - A095093(n).

cp's OEIS Frontend

A095355 Ratio A095106(n)/A095093(n) rounded down.

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A095356 Ratio A095106(n)/A095093(n) rounded to nearest integer.

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A095008 Number of 4k+3 primes (A002145) in range ]2^n,2^(n+1)].

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

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A095091 Number of 4k+3 integers in range [2^n, 2^(n+1)] whose Jacobi-vector is not a valid Motzkin-path (A095101).

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A095092 Number of 4k+3 primes whose Legendre-vector is a Dyck-path (A095102) in range ]2^n,2^(n+1)].

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