A001992 -id:A001992 - cp's OEIS frontend

A094852 Records in A001992.

5, 53, 173, 293, 2477, 9173, 61613, 74093, 170957, 360293, 679733, 2004917, 69009533, 138473837, 237536213, 384479933, 883597853, 1728061733, 9447241877, 49107823133, 1843103135837, 4316096218013, 15021875771117, 82409880589277, 326813126363093, 390894884910197, 1051212848890277

Offset: 1

N. J. A. Sloane, Jun 14 2004

nonn

Michel Marcus, Table of n, a(n) for n = 1..33 (all terms from N. J. A. Sloane)
Michael John Jacobson, Computational techniques in quadratic fields, Doctor of Science in Computer Science Thesis, University of Manitoba, 1995, 147 pages.
Michael John Jacobson Jr. and Hugh C. Williams, New quadratic polynomials with high densities of prime values, Math. Comp. 72 (2003), 499-519
D. H. Lehmer, E. Lehmer and D. Shanks, Integer sequences having prescribed quadratic character, Math. Comp., 24 (1970), 433-451.

Cf. A001992, A094853.

A094851 Class numbers associated with entries of A001992.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 5, 5, 3, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 9, 1, 1, 3, 3, 3, 3

Offset: 1

N. J. A. Sloane, Jun 14 2004

Michael John Jacobson, Computational techniques in quadratic fields, Doctor of Science in Computer Science Thesis, University of Manitoba, 1995, 147 pages.
Michael John Jacobson Jr. and Hugh C. Williams, New quadratic polynomials with high densities of prime values, Math. Comp. 72 (2003), 499-519
D. H. Lehmer, E. Lehmer and D. Shanks, Integer sequences having prescribed quadratic character, Math. Comp., 24 (1970), 433-451.

Cf. A001992.

A001986 Let p be the n-th odd prime. Then a(n) is the least prime congruent to 3 modulo 8 such that Legendre(-a(n), q) = -1 for all odd primes q <= p.

Original entry on oeis.org

19, 43, 43, 67, 67, 163, 163, 163, 163, 163, 163, 222643, 1333963, 1333963, 2404147, 2404147, 20950603, 51599563, 51599563, 96295483, 96295483, 146161723, 1408126003, 3341091163, 3341091163, 3341091163, 52947440683, 52947440683, 52947440683, 193310265163

Offset: 1

N. J. A. Sloane

nonn

Numbers so far are all congruent to 19 mod 24. - Ralf Stephan, Jul 07 2003
All terms are congruent to 19 mod 24. - Jianing Song, Feb 17 2019
Also a(n) is the least prime r congruent to 3 mod 8 such that the first n odd primes are quadratic nonresidues modulo r. Note that r == 3 (mod 8) implies 2 is a quadratic nonresidue modulo r. See A001992 for the case where r == 5 (mod 8). - Jianing Song, Feb 19 2019

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Jinyuan Wang, Table of n, a(n) for n = 1..56
Michael John Jacobson, Jr., Computational Techniques in Quadratic Fields, Master's thesis, University of Manitoba, Winnipeg, Manitoba, 1995.
Michael John Jacobson Jr. and Hugh C. Williams, New quadratic polynomials with high densities of prime values, Math. Comp. 72 (2003), 499-519.
D. H. Lehmer, E. Lehmer and D. Shanks, Integer sequences having prescribed quadratic character, Math. Comp., 24 (1970), 433-451.
D. H. Lehmer, E. Lehmer and D. Shanks, Integer sequences having prescribed quadratic character, Math. Comp., 24 (1970), 433-451 [Annotated scanned copy]

Cf. A001987, A094845, A094846.
Cf. A001992 (the congruent to 5 mod 8 case), A094851, A094852, A094853.
See A094841, A094842, A094843, A094844 for the case where the terms are not restricted to the primes.

isok(p, oddpn) = {forprime(q=3, oddpn, if (kronecker(-p, q) != -1, return (0));); return (1);}
a(n) = {my(oddpn = prime(n+1)); forprime(p=3, , if ((p%8) == 3, if (isok(p, oddpn), return (p));););} \\ Michel Marcus, Oct 17 2017

Revised by N. J. A. Sloane, Jun 14 2004

a(28)-a(30) from Jinyuan Wang, Apr 09 2020

A094847 Let p = n-th odd prime. Then a(n) = least positive integer congruent to 5 modulo 8 such that Legendre(a(n), q) = -1 for all odd primes q <= p.

Original entry on oeis.org

5, 53, 173, 173, 293, 437, 9173, 9173, 24653, 74093, 74093, 74093, 170957, 214037, 214037, 214037, 2004917, 44401013, 71148173, 154554077, 154554077, 163520117, 163520117, 163520117, 261153653, 261153653, 1728061733

Offset: 1

N. J. A. Sloane, Jun 14 2004

nonn

With an initial a(0) = 5, a(n) is the least fundamental discriminant D > 1 such that the first n + 1 primes are inert in the real quadratic field with discriminant D. See A094841 for the imaginary quadratic field case. - Jianing Song, Feb 15 2019

All terms are congruent to 5 mod 24. - Jianing Song, Feb 17 2019

Michael John Jacobson, Jr., Computational Techniques in Quadratic Fields, Master's thesis, University of Manitoba, Winnipeg, Manitoba, 1995.
Michael John Jacobson Jr. and Hugh C. Williams, New quadratic polynomials with high densities of prime values, Math. Comp. 72 (2003), 499-519.
D. H. Lehmer, E. Lehmer and D. Shanks, Integer sequences having prescribed quadratic character, Math. Comp., 24 (1970), 433-451.

Cf. A094848, A094849, A094850.
Cf. A094841 (the imaginary quadratic field case), A094842, A094843, A094844.
See A001992, A094851, A094852, A094853 for the case where the terms are restricted to the primes.

isok(m, oddpn) = {forprime(q=3, oddpn, if (kronecker(m, q) != -1, return (0));); return (1);}
a(n) = {oddpn = prime(n+1); m = 5; while(! isok(m, oddpn), m += 8); m;} \\ Michel Marcus, Oct 17 2017

A191089 Least prime p such that the first n primes are not squares mod p.

Original entry on oeis.org

3, 5, 43, 43, 67, 67, 163, 163, 163, 163, 163, 163, 74093, 170957, 360293, 679733, 2004917, 2004917, 51599563, 51599563, 96295483, 96295483, 146161723, 1408126003, 1728061733, 1728061733, 1728061733, 1728061733, 1728061733, 1728061733

Offset: 1

T. D. Noe, May 25 2011

nonn

That is, the first n primes are quadratic non-residues mod p. A less restrictive form of A001992. Sequence A191088 is similar, but allows p to be composite. See A147972 for the square version.

Cf. A001992, A147972, A191088.

Table[p = 2; While[Length[Select[Prime[Range[n]], JacobiSymbol[#, p] == -1 &]] < n, p = NextPrime[p]]; p, {n, 15}]

q=2;forprime(k=3,1e9,forprime(p=2,q,if(kronecker(p,k)>=0,next(2)));print1(k", ");q=nextprime(q+1);k--) \\ Charles R Greathouse IV, Oct 10 2011

a(16)-a(30) from Charles R Greathouse IV, Oct 10 2011

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A094852 Records in A001992.

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A094851 Class numbers associated with entries of A001992.

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A001986 Let p be the n-th odd prime. Then a(n) is the least prime congruent to 3 modulo 8 such that Legendre(-a(n), q) = -1 for all odd primes q <= p.

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A094847 Let p = n-th odd prime. Then a(n) = least positive integer congruent to 5 modulo 8 such that Legendre(a(n), q) = -1 for all odd primes q <= p.

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A191089 Least prime p such that the first n primes are not squares mod p.

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