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 1-7 of 7 results.

A112055 a(n) = A112054(n)/6.

Original entry on oeis.org

2, 3, 5, 7, 8, 12, 13, 15, 17, 18, 20, 22, 23, 27, 28, 30, 32, 33, 35, 37, 38, 40, 42, 43, 47, 48, 50, 52, 53, 55, 57, 58, 62, 63, 65, 67, 68, 70, 72, 73, 75, 77, 78, 82, 83, 85, 87, 88, 90, 92, 93, 97, 98, 100, 102, 103, 107, 108, 110, 112, 113, 117, 118, 120, 122
Offset: 1

Views

Author

Antti Karttunen, Aug 27 2005

Keywords

Crossrefs

Cf. A112054, A112082 (complement), A112085 (first differences).

Programs

  • Mathematica
    a112046[n_]:=Block[{i=1}, While[JacobiSymbol[i, 2n + 1]==1, i++]; i]; Select[Range[1000], a112046[2#] - a112046[2# - 1] != 0 &]/6 (* Indranil Ghosh, May 25 2017 *)
  • Python
    from sympy import jacobi_symbol as J
def a112046(n):
    i=1
    while True:
        if J(i, 2*n + 1)!=1: return i
        else: i+=1
def a(n): return a112046(2*n) - a112046(2*n - 1)
print([n//6 for n in range(1, 201) if a(n)!=0]) # Indranil Ghosh, May 25 2017

A112053 a(n) = A112046(2n) - A112046(2n-1) = A112048(n) - A112047(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, -6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
Offset: 1

Views

Author

Antti Karttunen, Aug 27 2005

Keywords

Links

Crossrefs

Cf. A112046, A112047, A112048.
Indices where a(n) is not zero: A112054. Values at those points: A112059.

Programs

  • Mathematica
    a112046[n_]:=Block[{i=1}, While[JacobiSymbol[i, 2n + 1]==1, i++]; i]; Table[a112046[2n] - a112046[2n - 1] , {n, 101}] (* Indranil Ghosh, May 24 2017 *)
  • Python
    from sympy import jacobi_symbol as J
def a112046(n):
    i=1
    while True:
        if J(i, 2*n + 1)!=1: return i
        else: i+=1
def a(n): return a112046(2*n) - a112046(2*n - 1)
print([a(n) for n in range(1, 102)]) # Indranil Ghosh, May 24 2017

Formula

a(n) = A112048(n)-A112047(n).

A112056 Odd numbers of the form 4n-1 for which Jacobi-first-non-one(4n-1) differs from Jacobi-first-non-one(4n+1).

Original entry on oeis.org

47, 71, 119, 167, 191, 287, 311, 359, 407, 431, 479, 527, 551, 647, 671, 719, 767, 791, 839, 887, 911, 959, 1007, 1031, 1127, 1151, 1199, 1247, 1271, 1319, 1367, 1391, 1487, 1511, 1559, 1607, 1631, 1679, 1727, 1751, 1799, 1847, 1871, 1967
Offset: 1

Views

Author

Antti Karttunen, Aug 27 2005

Keywords

Comments

Here Jacobi-first-non-one(m) (for odd numbers m) is defined as the first value of i >= 1, for which Jacobi symbol J(i,m) is not +1 (i.e. is either 0 or -1).

Crossrefs

Cf. A112054, A112057, A112058.

Programs

  • Mathematica
    a112046[n_]:=Block[{i=1}, While[JacobiSymbol[i, 2n + 1]==1, i++]; i]; 4*Select[Range[1000], a112046[2#] - a112046[2# - 1] != 0 &] - 1 (* Indranil Ghosh, May 24 2017 *)
  • Python
    from sympy import jacobi_symbol as J
def a112046(n):
    i=1
    while True:
        if J(i, 2*n + 1)!=1: return i
        else: i+=1
def a(n): return a112046(2*n) - a112046(2*n - 1)
print([4*n - 1 for n in range(1, 1001) if a(n)!=0]) # Indranil Ghosh, May 24 2017

Formula

a(n) = 4*A112054(n)-1.
a(n) = A112057(n)-2 = A112058(n)-1.

A112058 Mean of A112056 and A112057.

Original entry on oeis.org

48, 72, 120, 168, 192, 288, 312, 360, 408, 432, 480, 528, 552, 648, 672, 720, 768, 792, 840, 888, 912, 960, 1008, 1032, 1128, 1152, 1200, 1248, 1272, 1320, 1368, 1392, 1488, 1512, 1560, 1608, 1632, 1680, 1728, 1752, 1800, 1848, 1872, 1968
Offset: 1

Views

Author

Antti Karttunen, Aug 27 2005

Keywords

Crossrefs

a(n) = A112056(n)+1 = A112057(n)-1 = (A112056(n)+A112057(n))/2.

Programs

  • Mathematica
    a112046[n_]:=Block[{i=1}, While[JacobiSymbol[i, 2n + 1]==1, i++]; i]; 4*Select[Range[1000], a112046[2#] - a112046[2# - 1] != 0 &] (* Indranil Ghosh, May 24 2017 *)
  • Python
    from sympy import jacobi_symbol as J
def a112046(n):
    i=1
    while True:
        if J(i, 2*n + 1)!=1: return i
        else: i+=1
def a(n): return a112046(2*n) - a112046(2*n - 1)
print([4*n for n in range(1, 1001) if a(n)!=0]) # Indranil Ghosh, May 24 2017

Formula

a(n) = 4*A112054(n).

A112080 a(n) = difference between Jacobi-first-non-one(24n+1) and Jacobi-first-non-one(24n-1).

Original entry on oeis.org

0, 2, -2, 0, 4, 0, 8, -2, 0, 0, 0, 12, -6, 0, 12, 0, 2, -2, 0, -6, 0, 18, -6, 0, 0, 0, 2, -6, 0, -4, 0, 2, -2, 0, 18, 0, 2, -2, 0, 24, 0, 6, -2, 0, 0, 0, 6, -8, 0, 4, 0, 2, -2, 0, -6, 0, 32, -6, 0, 0, 0, 2, -6, 0, -10, 0, 2, -2, 0, 28, 0, 2, -2, 0, 4, 0, 38, -2, 0, 0, 0, 6, -6, 0, 4, 0, 2
Offset: 1

Views

Author

Antti Karttunen, Aug 27 2005

Keywords

Comments

The definition of Jacobi-first-non-one is given in A112056.

Links

Crossrefs

A112081(n) = a(n)/2. A112082 gives the points where a(n) is zero. A112059 gives the nonzero terms.

Formula

a(n) = 4*A112054(n)+1.

A112057 Odd numbers of the form 4n+1 for which Jacobi-first-non-one(4n-1) differs from Jacobi-first-non-one(4n+1).

Original entry on oeis.org

49, 73, 121, 169, 193, 289, 313, 361, 409, 433, 481, 529, 553, 649, 673, 721, 769, 793, 841, 889, 913, 961, 1009, 1033, 1129, 1153, 1201, 1249, 1273, 1321, 1369, 1393, 1489, 1513, 1561, 1609, 1633, 1681, 1729, 1753, 1801, 1849, 1873, 1969
Offset: 1

Views

Author

Antti Karttunen, Aug 27 2005

Keywords

Comments

The definition of Jacobi-first-non-one is given in A112056.

Crossrefs

a(n) = A112056(n)+2 = A112058(n)+1.

Programs

  • Mathematica
    a112046[n_]:=Block[{i=1}, While[JacobiSymbol[i, 2n + 1]==1, i++]; i]; 4*Select[Range[1000], a112046[2#] - a112046[2# - 1] != 0 &] + 1 (* Indranil Ghosh, May 25 2017 *)
  • PARI
    jfno(n) = my(k = 1); while(kronecker(k, n) == 1, k++); k;
lista(nn) = {forstep(n=5, nn, 4, if (jfno(n-2) != jfno(n), print1(n, ", ")););} \\ Michel Marcus, Jan 30 2018

Formula

a(n) = 4*A112054(n) + 1.

A112059 Nonzero terms of A112053 and A112080.

Original entry on oeis.org

2, -2, 4, 8, -2, 12, -6, 12, 2, -2, -6, 18, -6, 2, -6, -4, 2, -2, 18, 2, -2, 24, 6, -2, 6, -8, 4, 2, -2, -6, 32, -6, 2, -6, -10, 2, -2, 28, 2, -2, 4, 38, -2, 6, -6, 4, 2, -2, -4, 42, -6, 2, -8, -4, 2, -2, 2, -2, 6, 8, -2, 48, -6, 4, 2, -2, -10, 6, -12, 2, -6, -4, 2, -2, 2, -2, 52, 8
Offset: 1

Views

Author

Antti Karttunen, Aug 27 2005

Keywords

Links

Crossrefs

Cf. A112053, A112054, A112080.

Programs

  • Python
    from sympy import jacobi_symbol as J
def a112046(n):
    i=1
    while True:
        if J(i, 2*n + 1)!=1: return i
        else: i+=1
def a(n): return a112046(2*n) - a112046(2*n - 1)
print([a(n) for n in range(1, 201) if a(n)!=0]) # Indranil Ghosh, May 25 2017

Formula

a(n) = A112053(A112054(n)).
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.