A165580 -id:A165580 - cp's OEIS frontend

A165469 Numbers 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.

3, 7, 15, 23, 31, 39, 47, 55, 71, 79, 87, 95, 111, 119, 143, 151, 159, 167, 183, 191, 199, 215, 231, 239, 255, 263, 271, 287, 303, 311, 327, 335, 359, 383, 391, 399, 407, 431, 439, 447, 455, 471, 479, 503, 511, 519, 527, 535, 551, 559, 591, 599, 615, 623

Offset: 1

Antti Karttunen, Oct 06 2009

nonn

a(n) = A004767(A165468(n)). Setwise difference of A095100 and A165608. A165580 gives the primes in this sequence.

A165977 Primes 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, 59, 83, 103, 131, 251, 419, 563, 607, 659, 971, 1063, 1091, 1103, 1427, 1447, 1811, 1931, 1979, 2287, 2383, 2411, 2543, 2939, 3251, 3299, 3659, 3779, 3967, 4091, 4259, 4931, 5099, 5171, 5407, 5783, 5939, 6247, 6299, 6607, 6779, 6899

Offset: 1

Antti Karttunen, Oct 06 2009

nonn

Setwise difference of A095102 and A165580.

Subset of A165608.

Definition edited by Jonathan Sondow, Sep 26 2011

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

Jonathan Sondow, May 17 2010

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

			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.

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.

PlanetMath, Polya Vinogradov Inequality.
G. Polya, Über die Verteilung der quadratischen Reste und Nichtreste, Goettingen Nachr. (1918), 21-29.
I. Schur, Einige Bemerkungen zu der vorstehenden Arbeit des Herrn G. Polya: Über die Verteilung der quadratischen Reste und Nichtreste, Goettingen Nachr. (1918), pp. 30-36.
Eric Weisstein's World of Mathematics, Polya-Vinogradov Inequality.
Wikipedia, Character Sum.
Wikipedia, Legendre symbol.
Wikipedia, Quadratic residue.

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.

Table[Max[ Table[Abs[Sum[JacobiSymbol[i, Prime[n]], {i, 1, k}]], {k, 1, Prime[n] - 1}]], {n, 2, 100}]

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

a(n) = max_{01, and L(i/p) is the Legendre symbol.

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A165469 Numbers 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.

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A165977 Primes 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.

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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.

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