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 1-3 of 3 results.

A165608 Numbers of the form 4n+3 for which Sum_{i=1..u} J(i,4n+3) is never negative for any u in range [1,2n+1], but obtains at least one value zero, where J(i,m) is the Jacobi symbol.

Original entry on oeis.org

11, 27, 35, 59, 63, 75, 83, 103, 131, 135, 171, 175, 243, 251, 279, 295, 299, 319, 343, 351, 363, 371, 375, 395, 415, 419, 495, 539, 563, 567, 575, 607, 635, 659, 675, 703, 711, 731, 735, 755, 783, 867, 875, 895, 899, 903, 927, 943, 971, 999, 1063, 1067
Offset: 1

Views

Author

Antti Karttunen, Oct 06 2009

Keywords

Comments

Setwise difference of A095100 and A165469.

Crossrefs

a(n) = A004767(A165607(n)). Subset of A095100. A165977 gives the primes only. Cf. A165603.

Extensions

Definition edited by Jonathan Sondow, Sep 27 2011

A165580 Primes of the form 4n+3 for which Sum_{i=1..u} J(i,4n+3) is never zero for any u in range [1,(2n+1)], where J(i,m) is the Jacobi symbol.

Original entry on oeis.org

3, 7, 23, 31, 47, 71, 79, 151, 167, 191, 199, 239, 263, 271, 311, 359, 383, 431, 439, 479, 503, 599, 647, 719, 743, 751, 839, 863, 887, 911, 919, 983, 991, 1031, 1039, 1151, 1223, 1231, 1279, 1319, 1399, 1439, 1471, 1487, 1511, 1559, 1583, 1759
Offset: 1

Views

Author

Antti Karttunen, Oct 06 2009

Keywords

Crossrefs

Setwise difference of A095102 and A165977. Also subset of A165469.

A177865 Polya-Vinogradov numbers: a(n) is the maximum over all k > 0 of |#(quadratic residues modulo p up to k) - #(quadratic nonresidues modulo p up to k)| where p is the n-th prime and n > 1.

Original entry on oeis.org

1, 1, 2, 3, 2, 2, 3, 5, 3, 6, 4, 4, 5, 8, 5, 9, 4, 6, 10, 5, 10, 9, 6, 6, 7, 10, 9, 6, 6, 10, 15, 7, 9, 7, 14, 8, 9, 18, 9, 15, 7, 19, 8, 11, 18, 12, 15, 15, 9, 10, 22, 8, 21, 10, 21, 11, 22, 14, 10, 13, 11, 14, 25, 11, 13, 14, 12, 17, 12, 12, 27, 19, 16, 15, 27, 12, 12, 12, 12, 27, 11, 30
Offset: 2

Views

Author

Jonathan Sondow, May 17 2010

Keywords

Comments

In 1918 Polya and Vinogradov (independently) proved an upper bound for a(n) and Schur proved a lower bound:
sqrt(p)/2*pi < a(n) < sqrt(p)*log(p), where p is the n-th prime.
Named after the Hungarian mathematician George Pólya (1887-1985) and the Soviet mathematician Ivan Matveevich Vinogradov (1891-1983). - Amiram Eldar, Jun 22 2021

Examples

			The initial term is a(2) = 1 because the 2nd prime is 3 and L(1/3) = 1 and L(2/3) = -1, so |Sum_{i=1..k} L(i/3)| = 1 and 0 for k = 1 and 2, resp., and so max = 1.

References

  • I. M. Vinogradov, Über eine asymptotische Formel aus der Theorie der binaeren quadratischen Formen, J. Soc. Phys. Math. Univ. Permi, Vol. 1 (1918), pp. 18-28.
  • I. M. Vinogradov, Elements of Number Theory, 5th revised ed., Dover, 1954, p. 200.

Links

Crossrefs

Cf. A055088 (Triangle of generalized Legendre symbols L(a/b), with 1's for quadratic residues and 0's for quadratic non-residues [replace the 0's with -1's]).
Cf. also A095102, A165580, A165977, A207291, A207292.

Programs

  • Mathematica
    Table[Max[ Table[Abs[Sum[JacobiSymbol[i, Prime[n]], {i, 1, k}]], {k, 1, Prime[n] - 1}]], {n, 2, 100}]
  • PARI
    a(n) = my(p=prime(n)); vecmax(vector(p-1, k, vecsum(vector(k, i, issquare(Mod(i, p)))) - vecsum(vector(k, i, !issquare(Mod(i, p)))))); \\ Michel Marcus, Mar 03 2023

Formula

a(n) = max_{01, and L(i/p) is the Legendre symbol.
Showing 1-3 of 3 results.

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

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