cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A095093 Number of 4k+3 primes whose Legendre-vector is not Dyck-path (A095103) in range ]2^n,2^(n+1)].

Original entry on oeis.org

0, 0, 0, 1, 1, 3, 6, 12, 21, 41, 77, 143, 287, 530, 1010, 1967, 3711, 7125, 13806, 26525, 51126
Offset: 1

Views

Author

Antti Karttunen, Jun 01 2004

Keywords

Links

Crossrefs

Cf. A095103.

Programs

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

Formula

a(n) = A095008(n) - A095092(n).

A095273 a(n) = (A095103(n)-3)/4.

Original entry on oeis.org

4, 10, 16, 26, 31, 34, 40, 44, 52, 55, 56, 70, 76, 82, 86, 91, 94, 110, 115, 116, 121, 122, 124, 130, 136, 142, 146, 154, 157, 160, 170, 172, 181, 184, 196, 202, 205, 206, 214, 220, 226, 236, 241, 254, 262, 271, 280, 290, 292, 296, 314, 320, 322
Offset: 1

Views

Author

Antti Karttunen, Jun 01 2004

Keywords

Links

Crossrefs

Complement of A095272 in A095278, subset of A095275.

A095108 Diving index of the n-th diving 4k+3 prime (A095103(n)).

Original entry on oeis.org

3, 3, 3, 7, 7, 3, 3, 11, 3, 13, 61, 3, 3, 3, 7, 45, 3, 7, 7, 35, 7, 35, 3, 3, 3, 3, 15, 3, 15, 3, 7, 3, 45, 3, 3, 3, 7, 7, 3, 3, 3, 7, 7, 67, 3, 7, 3, 7, 3, 7, 57, 7, 3, 7, 15, 7, 45, 7, 23, 3, 3, 11, 7, 3, 89, 13, 55, 3, 45, 35, 3, 7, 7, 3, 3, 3, 13, 7, 15, 3, 19, 3, 7, 3, 7, 3, 99
Offset: 1

Views

Author

Antti Karttunen, Jun 01 2004

Keywords

Links

Crossrefs

Nonzero terms of A095104. Cf. A095271.

A095102 Odd primes p for which all sums Sum_{i=1..u} L(i/p) (with u ranging from 1 to (p-1)) are nonnegative, where L(i/p) is Legendre symbol of i and p, defined to be 1 if i is a quadratic residue (mod p) and -1 if i is a quadratic non-residue (mod p).

Original entry on oeis.org

3, 7, 11, 23, 31, 47, 59, 71, 79, 83, 103, 131, 151, 167, 191, 199, 239, 251, 263, 271, 311, 359, 383, 419, 431, 439, 479, 503, 563, 599, 607, 647, 659, 719, 743, 751, 839, 863, 887, 911, 919, 971, 983, 991, 1031, 1039, 1063, 1091, 1103, 1151
Offset: 1

Views

Author

Antti Karttunen, Jun 01 2004

Keywords

Comments

All 4k+3 primes whose Legendre-vector (cf. A055094) forms a valid Dyck-path (cf. A014486).

Links

Crossrefs

Intersection of A000040 and A095100. Subset of A080114 (see comments there). Complement of A095103 in A002145.
Cf. A095092.

Programs

  • Mathematica
    isMotzkin[n_, k_] := Module[{s = 0, r = True}, Do[s += JacobiSymbol[i, n]; If[s < 0, r = False; Break[]], {i, 1, k}]; r]; A095102[max_] := Select[ Range[3, max, 4], PrimeQ[#] && isMotzkin[#, Quotient[#, 2]]&]; A095102[1151] (* Jean-François Alcover, Feb 16 2018, after Peter Luschny *)
  • PARI
    isok(m) = {if(!isprime(m-(m<3)), return(0)); my(s=0); for(i=1, m-1, if((s+=kronecker(i, m))<0, return(0))); 1; } \\ Jinyuan Wang, Jul 20 2020
  • Sage
    def A095102_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(3, n+1, 4))
    return filter(lambda m: is_Motzkin(m, m//2), P)
A095102_list(1151) # Peter Luschny, Aug 09 2012

Formula

a(n) = 4*A095272(n) + 3.

A095101 Integers m of the form 4k+3 for which some of the sums Sum_{i=1..u} J(i/m) (with u ranging from 1 to (m-1)) is negative, where J(i/m) is Jacobi symbol of i and m.

Original entry on oeis.org

19, 43, 51, 67, 91, 99, 107, 115, 123, 127, 139, 147, 155, 163, 179, 187, 195, 203, 207, 211, 219, 223, 227, 235, 247, 259, 267, 275, 283, 291, 307, 315, 323, 331, 339, 347, 355, 367, 379, 387, 403, 411, 423, 427, 435, 443, 451, 459, 463, 467
Offset: 1

Views

Author

Antti Karttunen, Jun 01 2004

Keywords

Comments

Integers whose Jacobi-vector does not form a valid Motzkin-path.

Links

Crossrefs

Subset of A095103. Complement of A095100 in A004767.
Cf. A095091.

Programs

  • PARI
    isok(m) = {my(s=0); if(m%4==3, for(i=1, m-1, if((s+=kronecker(i, m))<0, return(1)))); 0; } \\ Jinyuan Wang, Jul 20 2020
  • Sage
    def is_Motzkin(n, k):
    s = 0
    for i in (1..k) :
        s += jacobi_symbol(i, n)
        if s < 0 : return False
    return True
def A095101_list(n):
    return [m for m in range(3, n+1, 4) if not is_Motzkin(m, m//2)]
A095101_list(467) # Peter Luschny, Aug 08 2012

Formula

a(n) = 4*A095275(n) + 3.

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

Original entry on oeis.org

0, 0, 0, 3, 3, 5, 15, 13, 10, 13, 25, 22, 97, 85, 203, 359, 625, 1067, 1880, 3166, 6068
Offset: 1

Views

Author

Antti Karttunen, Jun 12 2004

Keywords

Comments

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.

Crossrefs

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

Formula

a(n) = 0 if A095093(n) is 0, otherwise a(n) = floor(A095106(n)/A095093(n)).
Showing 1-6 of 6 results.

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