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 1-9 of 9 results.

A165472 Partial sums of A165471.

Original entry on oeis.org

0, 1, 2, 1, 2, 1, 0, -1, 0, 1, 0, -1, -2, -1, -2, -1, 0, 1, 2, 3, 2, 3, 2, 1, 0, 1, 2, 1, 0, -1, 0, -1, 0, 1, 2, 3, 4, 5, 6, 5, 4, 3, 4, 3, 2, 1, 0, -1, -2, -1, 0, -1, 0, 1, 0, 1, 0, -1, -2, -3, -2, -3, -4, -5, -4, -5, -4, -5, -4, -3, -2, -1, 0, -1, 0, -1, 0, 1, 0, 1, 0, 1, 0, -1, 0, -1, -2
Offset: 0

Views

Author

Antti Karttunen, Sep 21 2009

Keywords

Comments

Period 65537.
Maximum value is 160 at a(65537n + 16382) and a(65537n + 16384); minimum value is -160 at a(65537n + 49152) and a(65537n + 49154). - Charles R Greathouse IV, Aug 06 2012

Links

Crossrefs

A165486 gives the squared version. Positions of zeros: A165473. See also A165474, A165475. Compare also to A165477, A165482.

A112049 a(n) = position of A112046(n) in A000040.

Original entry on oeis.org

1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 3, 3, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 3, 4, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 4, 3, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 3, 3, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 4, 5, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 3, 3, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 3, 6, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 4, 3, 1, 1, 2, 2, 1, 1
Offset: 1

Views

Author

Antti Karttunen, Aug 27 2005

Keywords

Comments

A112051 gives the first positions of distinct new values in this sequence, that seem also to be the positions of the first occurrence of each n, and thus the positions of the records. Compare also to A084921. - Antti Karttunen, May 26 2017

Links

Crossrefs

Cf. A049084, A112046, A112051, A112060, A165471, A268829, A286465, A286466.
Cf. A286579 (ordinal transform).

Programs

  • Mathematica
    a112046[n_]:=Block[{i=1},While[JacobiSymbol[i, 2n + 1]==1, i++]; i];a049084[n_]:=If[PrimeQ[n], PrimePi[n], 0]; Table[a049084[a112046[n]], {n, 102}] (* Indranil Ghosh, May 11 2017 *)
  • PARI
    A112049(n) = for(i=1, (2*n), if((kronecker(i, (n+n+1)) < 1), return(primepi(i)))); \\ Antti Karttunen, May 26 2017
  • Python
    from sympy import jacobi_symbol as J, isprime, primepi
def a049084(n):
    return primepi(n) if isprime(n) else 0
def a112046(n):
    i=1
    while True:
        if J(i, 2*n + 1)!=1: return i
        else: i+=1
def a(n): return a049084(a112046(n))
print([a(n) for n in range(1, 103)]) # Indranil Ghosh, May 11 2017

Formula

a(n) = A049084(A112046(n)).

Extensions

Unnecessary fallback-clause removed from the name by Antti Karttunen, May 26 2017

A268819 Column 32769 of array A269158: a(n) = F(n,65537), function F as defined in A269158.

Original entry on oeis.org

0, 98305, 3, 0, 6, 98306, 2, 98305, 12, 98311, 14, 3, 1, 98307, 9, 0, 24, 98317, 24, 6, 16, 98319, 27, 98306, 0, 98304, 23, 2, 30, 98312, 2, 98305, 48, 98329, 0, 12, 52, 98329, 6, 98311, 3, 98321, 3, 14, 14, 98330, 3, 3, 41, 98305, 43, 1, 4, 98326, 45, 98307, 6, 98335, 43, 9, 27, 98307, 19, 0, 27, 98353, 2, 24, 100, 98305, 1
Offset: 1

Views

Author

Antti Karttunen, Feb 25 2016

Keywords

Comments

Terms a(1) .. a(65536) occur as column 32769 in arrays A268728 and A269158.

Links

Crossrefs

Cf. A010873, A052928, A165471, A165472.
Cf. arrays A268728 and A269158.
Cf. A269157 (indices of zeros).

Programs

  • Scheme
    ;; Two variants, both give same results in range n=1..65536:
(define (A268819 n) (A268728auxbi n 65537))
(define (A268819 n) (A269158auxbi n 65537))

Formula

a(n) = F(n,65537) = A269158(n,32769), function F as defined in A269158.
Other identities. For n = 1..65536:
a(n) = A268728(n,32769).
A165471(n) = 1 - A010873(A052928(a(n))).

A165591 Jacobi symbol (n,59701).

Original entry on oeis.org

0, 1, -1, 1, 1, 1, -1, -1, -1, 1, -1, 1, 1, -1, 1, 1, 1, -1, -1, -1, 1, -1, -1, 1, -1, 1, 1, 1, -1, -1, -1, -1, -1, 1, 1, -1, 1, -1, 1, -1, -1, 1, 1, 1, 1, 1, -1, -1, 1, 1, -1, -1, -1, -1, -1, 1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, -1, 1, -1, 1, 1, -1, -1, -1, 1, 1, -1, -1, 1, -1, 1
Offset: 0

Views

Author

Antti Karttunen, Sep 22 2009

Keywords

Comments

Semiprime 59701 = 227*263 = A005385(11)*A005385(12).

Links

Crossrefs

a(n) = A011626(n)*A165574(n). Partial sums: A165592. Cf. A165471.

A179417 a(n) is the binary number (shown here in decimal) constructed from quadratic residues of 65537 in range [(n^2)+1,(n+1)^2] in such a way that quadratic residues are mapped to 1-bits, and non-quadratic residues (as well as the multiples of 65537) to 0-bits, with the lower end of range mapped to less significant, and the higher end of range to more significant bits.

Original entry on oeis.org

1, 5, 24, 104, 279, 2001, 4131, 17453, 88826, 362532, 1655660, 6120642, 25376649, 128526482, 301370205, 1756488602, 8046359747, 30854867177, 73845140753, 488906501177, 2106640948770, 6573967883049, 29711211505300
Offset: 0

Views

Author

Antti Karttunen, Jul 27 2010

Keywords

Comments

The binary width of terms are 1, 3, 5, 7, 9, ... i.e., the successive odd numbers, as their partial sums give the squares, 1, 4, 9, 16, ... at which points there certainly is always a quadratic residue, which thus gives the most significant bit for each number.

Examples

			In the range [(2^2)+1, (2+1)^2] (i.e., [5,9]) we have A165471(5)=A165471(6)=A165471(7)=-1 and A165471(8)=A165471(9)=+1, i.e., there are quadratic non-residues at points 5, 6 and 7, and quadratic residues at 8 and 9, so we construct a binary number 11000, which is 24 in decimal, thus a(2)=24.

Links

Crossrefs

Cf. A179418.
Compare to similar bit triangle illustrations given in A080070, A122229, A122232, A122235, A122239, A122242, A122245.

A165596 Jacobi symbol (n,59881).

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, -1, 1, 1, 1, -1, 1, 1, -1, 1, 1, -1, 1, -1, 1, -1, -1, 1, 1, 1, 1, 1, -1, 1, 1, 1, 1, -1, -1, -1, 1, -1, -1, 1, 1, 1, -1, 1, -1, 1, 1, 1, 1, 1, 1, -1, 1, 1, 1, -1, -1, -1, 1, -1, 1, -1, 1, -1, 1, 1, -1, -1, -1, 1, -1, -1, 1, -1, -1, 1, -1, 1, 1, -1, 1, 1, 1, 1, -1, -1
Offset: 0

Views

Author

Antti Karttunen, Sep 22 2009

Keywords

Comments

Semiprime 59881 = 233*257 = A005478(6)*A019434(3) = A117879(11).

Links

Crossrefs

a(n) = A011628(n)*A165573(n). Partial sums: A165597. Cf. A165591, A165471.

A179418 a(n) = Number of quadratic residues of 65537 in range [(n^2)+1,(n+1)^2].

Original entry on oeis.org

1, 2, 2, 3, 5, 7, 4, 6, 11, 6, 10, 11, 12, 12, 17, 16, 20, 21, 15, 19, 24, 24, 21, 22, 24, 29, 25, 25, 30, 29, 37, 29, 34, 30, 31, 36, 37, 34, 32, 36, 46, 43, 46, 50, 46, 48, 39, 40, 52, 49, 55, 51, 54, 47, 55, 50, 53, 65, 57, 59, 74, 73, 78, 65, 66, 70, 58, 64, 69, 69, 69, 78
Offset: 0

Views

Author

Antti Karttunen, Jul 27 2010

Keywords

Comments

Alternatively, a number of 1-bits in A179417(n). Sum_{i=0..255} a(i) = 32768. See also A165471.

Links

Formula

a(n) = A000120(A179417(n))

A179416 a(n)=1 if (n modulo 65536)+1 is a quadratic residue of 65537, 0 otherwise.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0
Offset: 0

Views

Author

Antti Karttunen, Jul 27 2010

Keywords

Comments

This sequence gives essentially the same information as A165471, but in contrast to it (and A165472), the period of this sequence is explicitly defined as 65536 (instead of 65537), so that in essence the zeros at A165471(k*65537) are silently skipped. Several derived sequences to be computed.

Links

Programs

  • Sage
    def A179416_list(n) :  # for n <= 65536
    Q = quadratic_residues(65537)
    return [int(i in Q) for i in (1..n)]
A179416_list(102) # Peter Luschny, Aug 08 2012

Formula

a(n) = 1 if A165471(1+(n%65536))=+1, otherwise 0. Period 65536.

A290108 a(n) = A268819(n) mod 8.

Original entry on oeis.org

0, 1, 3, 0, 6, 2, 2, 1, 4, 7, 6, 3, 1, 3, 1, 0, 0, 5, 0, 6, 0, 7, 3, 2, 0, 0, 7, 2, 6, 0, 2, 1, 0, 1, 0, 4, 4, 1, 6, 7, 3, 1, 3, 6, 6, 2, 3, 3, 1, 1, 3, 1, 4, 6, 5, 3, 6, 7, 3, 1, 3, 3, 3, 0, 3, 1, 2, 0, 4, 1, 1, 5, 6, 5, 7, 0, 0, 7, 4, 6, 0, 2, 7, 0, 6, 2, 1, 7, 3, 7, 2, 3, 5, 2, 2, 2, 1, 0, 2, 0, 3, 2, 4, 0, 3, 5, 0, 7, 4, 4, 6, 2, 2, 7, 0, 6, 1, 2, 2, 0
Offset: 1

Views

Author

Antti Karttunen, Sep 11 2017

Keywords

Links

Crossrefs

Cf. A010877, A165471, A268728, A268819, A269157.

Programs

Formula

a(n) = A010877(A268819(n)) = A268819(n) mod 8.
Showing 1-9 of 9 results.

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