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 1-5 of 5 results.

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)).

A095360 Ratio A095109(n)/A095091(n) rounded to nearest integer.

Original entry on oeis.org

0, 0, 0, 3, 7, 6, 13, 13, 21, 25, 43, 59, 98, 141, 287, 479, 841, 1504, 2789, 5049
Offset: 1

Views

Author

Antti Karttunen, Jun 12 2004

Keywords

Crossrefs

See Comments on A095359, where the same ratios are given rounded down.

Formula

a(n) = 0 if A095091(n) is 0, otherwise a(n) = round(A095109(n)/A095091(n)).

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.

A095090 Number of 4k+3 integers in range ]2^n,2^(n+1)] whose Jacobi-vector is a Motzkin-path (A095100).

Original entry on oeis.org

1, 1, 2, 3, 6, 9, 17, 33, 60, 108, 202, 360, 703, 1328, 2519, 4779, 9103, 17501, 33473, 64761
Offset: 1

Views

Author

Antti Karttunen and Jun 01 2004

Keywords

Links

Crossrefs

Cf. A095092, A095100.

Programs

  • PARI
    is(m) = {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

Formula

a(n) = 2^(n-2) - A095091(n) for n > 1.

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 1-5 of 5 results.

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