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.

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

Views

Author

Antti Karttunen, Jun 01 2004

Keywords

Crossrefs

Cf. A095093.

Formula

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

A095275 a(n) = (A095101(n)-3)/4.

Original entry on oeis.org

4, 10, 12, 16, 22, 24, 26, 28, 30, 31, 34, 36, 38, 40, 44, 46, 48, 50, 51, 52, 54, 55, 56, 58, 61, 64, 66, 68, 70, 72, 76, 78, 80, 82, 84, 86, 88, 91, 94, 96, 100, 102, 105, 106, 108, 110, 112, 114, 115, 116, 118, 120, 121, 122, 124, 126, 128, 130, 132
Offset: 0

Views

Author

Antti Karttunen, Jun 01 2004

Keywords

Crossrefs

Cf. A095101. Complement of A095274. Subset: A095273.

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

Views

Author

Antti Karttunen, Jun 01 2004

Keywords

Crossrefs

Formula

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

A095100 Integers m of the form 4k+3 for which all sums Sum_{i=1..u} J(i/m) (with u ranging from 1 to (m-1)) are nonnegative, where J(i/m) is Jacobi symbol of i and m.

Original entry on oeis.org

3, 7, 11, 15, 23, 27, 31, 35, 39, 47, 55, 59, 63, 71, 75, 79, 83, 87, 95, 103, 111, 119, 131, 135, 143, 151, 159, 167, 171, 175, 183, 191, 199, 215, 231, 239, 243, 251, 255, 263, 271, 279, 287, 295, 299, 303, 311, 319, 327, 335, 343, 351, 359, 363
Offset: 1

Views

Author

Antti Karttunen, Jun 01 2004

Keywords

Comments

Integers whose Jacobi-vector forms a valid Motzkin-path.

Crossrefs

Subset of A095102. Complement of A095101 in A004767.
Cf. A095090.

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]; A095100[n_] := Select[4*Range[0, n+1]+3, isMotzkin[#, Quotient[#, 2]] &]; A095100[90] (* Jean-François Alcover, Oct 08 2013, translated from Sage *)
  • PARI
    isok(m) = {if(m%4<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 is_Motzkin(n, k):
        s = 0
        for i in range(1, k + 1) :
            s += jacobi_symbol(i, n)
            if s < 0: return False
        return True
    def A095100_list(n):
        return [m for m in range(3, n + 1, 4) if is_Motzkin(m, m // 2)]
    A095100_list(363) # Peter Luschny, Aug 08 2012
    

Formula

a(n) = 4*A095274(n) + 3.

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

Views

Author

Antti Karttunen, Jun 01 2004

Keywords

Crossrefs

Intersection of A000040 and A095101. Complement of A095102 in A002145.
Cf. A095093, A095108 (diving indices).

Programs

  • Mathematica
    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 *)
  • 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
    
  • Sage
    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

Formula

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

A095359 Ratio A095109(n)/A095091(n) rounded down.

Original entry on oeis.org

0, 0, 0, 3, 6, 5, 12, 12, 20, 24, 42, 58, 97, 140, 286, 478, 841, 1504, 2788, 5048
Offset: 1

Views

Author

Antti Karttunen, Jun 12 2004

Keywords

Comments

This is the average diving index for those 4k+3 integers in range ]2^n,2^(n+1)] that "dive". See A095101.
The ratios before rounding are: 0, 0, 0, 3, 6.5, 5.714286, 12.933333, 12.548387, 20.691176, 24.635135, 42.903226, 58.98494, 97.742751, 140.742413, 286.896704, 478.786471, 841.487894, 1504.108692, 2788.84881, 5048.608416.

Crossrefs

A095360 gives the same ratios rounded to nearest integer. A095355 gives similar ratios computed only for 4k+3 primes.

Formula

a(n) = 0 if A095091(n) is 0, otherwise a(n) = floor(A095109(n)/A095091(n)).
Showing 1-6 of 6 results.