A053760 -id:A053760 - cp's OEIS frontend

A177864 a(n) is the smallest nontrivial quadratic residue modulo prime(n), for n >= 3.

4, 2, 3, 3, 2, 4, 2, 4, 2, 3, 2, 4, 2, 4, 3, 3, 4, 2, 2, 2, 3, 2, 2, 4, 2, 3, 3, 2, 2, 3, 2, 4, 4, 2, 3, 4, 2, 4, 3, 3, 2, 2, 4, 2, 4, 2, 3, 3, 2, 2, 2, 3, 2, 2, 4, 2, 3, 2, 4, 4, 4, 2, 2, 4, 4, 2, 3, 3, 2, 2, 2, 3, 4, 2, 4, 3, 2, 2, 3, 3, 2, 2, 2, 3, 2, 2, 4, 2, 3, 2, 2, 3, 4, 2, 4, 2, 4, 3

Offset: 3

Jonathan Sondow, May 16 2010

There is no quadratic residue > 1 modulo the first or 2nd prime, so the sequence begins with a(3).

			The quadratic residues modulo prime(3) = 5 are 1 and 4, so a(3) = 4.

Wikipedia, Quadratic residue
Wikipedia, Quadratic reciprocity

Cf. A063987 (triangle in which the n-th row gives the quadratic residues modulo prime(n)), A053760 (smallest positive quadratic nonresidue modulo prime(n)).

Flatten[Table[ Extract[Flatten[ Position[Table[JacobiSymbol[i, Prime[n]], {i, 1, Prime[n] - 1}], 1]], {2}], {n, 3, 100}]]

a(n,p=prime(n))=[2,0,0,0,4,0,2,0,0,0,3,0,3,0,0,0,2,0,4,0,0,0,2][p%24] \\ Charles R Greathouse IV, Jun 14 2022

a(n) = 2 or 3 or 4 according as prime(n) == 1,7,9,15,17,23 or 11,13 or 3,5,19,21 (mod 24), respectively, for n > 2, by the quadratic reciprocity law and its supplements.

A225225 Second smallest prime quadratic non-residue modulo the n-th prime.

Original entry on oeis.org

3, 3, 3, 5, 7, 5, 5, 3, 7, 3, 11, 5, 7, 3, 11, 3, 11, 7, 3, 11, 7, 7, 5, 7, 7, 3, 5, 5, 11, 5, 5, 17, 5, 3, 3, 7, 5, 3, 13, 3, 7, 7, 11, 11, 3, 11, 3, 5, 5, 7, 5, 13, 11, 11, 5, 7, 3, 13, 5, 11, 3, 3, 3, 17, 7, 3, 3, 11, 5, 7, 5, 13, 5, 5, 3, 11, 3, 5, 13, 11, 11, 13, 13, 7, 17, 5, 13, 11, 3, 5, 5, 17, 5, 7, 3, 17, 3, 7, 3, 11

Offset: 1

Jonathan Sondow, May 02 2013

nonn

See comments, references and links in A053760 = Smallest positive quadratic nonresidue modulo the n-th prime.

			The positive quadratic non-residues modulo prime(5) = 11 are 2, 6, 7, 8, 10, 11, ... and the second smallest prime among them is 7, so a(5) = 7.

Kevin J. McGown, On the second smallest prime non-residue, J. Number Theory, 133 (2013), 1289-1299.

Cf. A053760.

Table[p = Prime[n]; Part[ Select[ Range[p+1], JacobiSymbol[#, p] != 1 &], 2], {n, 1, 100}]

A249271 Decimal expansion of the mean value over all positive integers of a function giving the least quadratic nonresidue modulo a given odd integer (this function is precisely defined in A053761).

Original entry on oeis.org

3, 1, 4, 7, 7, 5, 5, 1, 4, 8, 5, 0, 2, 4, 0, 0, 3, 1, 2, 5, 1, 6, 6, 7, 4, 9, 5, 5, 8, 7, 9, 7, 6, 9, 2, 0, 9, 2, 7, 2, 9, 3, 7, 7, 4, 8, 7, 9, 3, 3, 9, 8, 8, 6, 4, 0, 5, 9, 6, 4, 7, 0, 2, 0, 6, 6, 4, 7, 8, 1, 1, 8, 0, 0, 9, 1, 6, 7, 2, 4, 6, 7, 7, 9, 9, 7, 9, 4, 5, 2, 0, 9, 4, 8, 8, 2, 8, 7, 9, 7, 8, 6, 9, 1

Offset: 1

Jean-François Alcover, Oct 24 2014

			3.147755148502400312516674955879769209272937748793398864...

Steven R. Finch, Mathematical Constants, Cambridge Univ. Press, 2003, Meissel-Mertens constants: Quadratic residues, pp. 96—98.

Robert Baillie and Samuel S. Wagstaff, Lucas pseudoprimes, Mathematics of Computation 35 (1980), pp. 1391-1417.

Cf. A053760, A053761, A098990.

digits = 104; Clear[s]; s[m_] := s[m] = 1 + Sum[(Prime[j] + 1)*2^(-j + 1)* Product[1 - 1/Prime[i], {i, 1, j - 1}] // N[#, digits + 100]&, {j, 2, m}] ; s[10]; s[m = 20]; While[RealDigits[s[m]] != RealDigits[s[m/2]], Print[m]; m = 2*m]; RealDigits[s[m], 10, digits] // First

do(lim)=my(p=2,pr=1.,s=1); forprime(q=3,lim, pr*=(1-1/p)/2; s+=(q+1)*pr; p=q); s \\ Charles R Greathouse IV, Dec 20 2017

1 + sum_{j=2..m} (p_j + 1)*2^(-j+1)*prod_{i=1..j-1} (1 - 1/p_i), where p_j is the j-th prime number.

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A177864 a(n) is the smallest nontrivial quadratic residue modulo prime(n), for n >= 3.

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A225225 Second smallest prime quadratic non-residue modulo the n-th prime.

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A249271 Decimal expansion of the mean value over all positive integers of a function giving the least quadratic nonresidue modulo a given odd integer (this function is precisely defined in A053761).

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