A094847 -id:A094847 - cp's OEIS frontend

A094849 Records in A094847.

5, 53, 173, 293, 437, 9173, 24653, 74093, 170957, 214037, 2004917, 44401013, 71148173, 154554077, 163520117, 261153653, 1728061733, 9447241877, 19553206613, 49107823133, 385995595277, 13213747959653, 14506773263237, 57824199003317, 160909740894437, 370095509388197, 1409029796180597, 4075316253649373, 18974003020179917, 224117990614052477, 637754768063384837, 4472988326827347533

Offset: 1

N. J. A. Sloane, Jun 14 2004

nonn

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. A094847, A094848, A094850.

A094848 Class numbers associated with entries in A094847.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 4, 5, 5, 5, 3, 2, 2, 2, 3, 2, 6, 2, 2, 9, 9, 9, 3, 3, 1, 1, 1, 7, 2, 1, 1, 2, 2, 2, 2, 1, 1, 4, 4, 4, 4, 2, 1, 1, 4, 4, 4, 4, 4, 3, 3, 3, 12, 12, 12, 12, 12

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. A094847.

A094841 Let p = n-th odd prime. Then a(n) = least positive integer 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, 77683, 77683, 1333963, 2404147, 2404147, 20950603, 36254563, 51599563, 96295483, 96295483, 114148483, 269497867, 269497867, 269497867, 269497867, 585811843, 52947440683

Offset: 1

N. J. A. Sloane, Jun 13 2004

nonn

(a(n-1) + 1)/4 is the least positive integer c such that x^2 + x + c is not divisible by the first n primes. This implies that a(n) is congruent to 19 mod 24 and that a(n) is congruent to 43 or 67 mod 120 for n > 1. - William P. Orrick, Mar 19 2017

With an initial a(0) = 3, a(n) is the negated fundamental discriminant D < 0 with the least absolute value such that the first n + 1 primes are inert in the imaginary quadratic field with discriminant D. See A094847 for the real discriminant case. - Jianing Song, Feb 15 2019

William P. Orrick, Table of n, a(n) for n = 1..58 (first 28 terms from N. J. A. Sloane)
M. J. Jacobson, Jr., Computational Techniques in Quadratic Fields, Master's thesis, University of Manitoba, Winnipeg, Manitoba, 1995. (This sequence is given in Table 6.6.)
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. A094842, A094843, A094844.
Cf. A094847 (the real quadratic field case), A094848, A094849, A094850.
See A001986, A001987, A094845, A094846 for the case where the terms are restricted to the primes.
Cf. also A181667.

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

a(n) = 4*A181667(n+1) - 1. - William P. Orrick, Mar 19 2017

A001992 Let p = n-th odd prime. Then a(n) = least prime 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, 2477, 9173, 9173, 61613, 74093, 74093, 74093, 170957, 360293, 679733, 2004917, 2004917, 69009533, 138473837, 237536213, 384479933, 883597853, 1728061733, 1728061733, 1728061733, 1728061733

Offset: 1

N. J. A. Sloane

nonn

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

Also a(n) is the least prime r congruent to 5 mod 8 such that the first n odd primes are quadratic nonresidues modulo r. Note that r == 5 (mod 8) implies 2 is a quadratic nonresidue modulo r. See A001986 for the case where r == 3 (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).

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. [There is an error in the table given in this paper.]
D. H. Lehmer, E. Lehmer and D. Shanks, Integer sequences having prescribed quadratic character, Math. Comp., 24 (1970), 433-451 [Annotated scanned copy]

Cf. A094851, A094852, A094853.
Cf. A001986 (the congruent to 3 mod 8 case), A001987, A094845, A094846.
See A094847, A094848, A094849, A094850 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) == 5, if (isok(p, oddpn), return (p));););} \\ Michel Marcus, Oct 17 2017

Corrected and extended by N. J. A. Sloane, Jun 14 2004

A241482 Least fundamental discriminant D > 1 such that the first n primes p have (D/p) >= 0.

Original entry on oeis.org

8, 12, 24, 60, 60, 364, 984, 1596, 1596, 1596, 3705, 58444, 84396, 164620, 172236, 369105, 369105, 731676, 731676, 3442296, 3442296, 32169916, 32169916, 47973864, 47973864, 47973864, 313114620, 313114620, 313114620, 313114620, 13461106065, 27765196680, 40527839121, 55213498824, 55213498824, 381031123720

Offset: 1

Charles R Greathouse IV, Apr 23 2014

nonn

By the Chinese Remainder Theorem and Prime Number Theorem in arithmetic progressions, this sequence is infinite.

a(n) is the least fundamental discriminant D > 1 such that the first n primes either decompose or ramify in the real quadratic field with discriminant D. See A306218 for the imaginary quadratic field case. - Jianing Song, Feb 14 2019

			(364/2) = 0, (364/3) = 1, (364/5) = 1, (364/7) = 0, (364/11) = 1, (364/13) = 0, so 3, 5, 11 decompose in Q[sqrt(91)] and 2, 7, 13 ramify in Q[sqrt(-231)]. For other fundamental discriminants 1 < D < 364, at least one of 2, 3, 5, 7, 11, 13 is inert in the imaginary quadratic field with discriminant D, so a(6) = 364. - _Jianing Song_, Feb 14 2019

Cf. A003658, A232931, A306218 (the imaginary quadratic field case).

A002189 and A094847 are similar sequences.

a(n) = my(i=2); while(!isfundamental(i)||sum(j=1, n, kronecker(i,prime(j))==-1)!=0, i++); i \\ Jianing Song, Feb 14 2019

a(n) > prime(n)^(4*sqrt(e) + o(1)). - Charles R Greathouse IV, Apr 23 2014

a(n) = A003658(k), where k is the smallest number such that A232931(k) >= prime(n+1). - Jianing Song, Feb 15 2019

a(36) from Charles R Greathouse IV, Apr 24 2014

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A094849 Records in A094847.

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A094848 Class numbers associated with entries in A094847.

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A094841 Let p = n-th odd prime. Then a(n) = least positive integer congruent to 3 modulo 8 such that Legendre(-a(n), q) = -1 for all odd primes q <= p.

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A001992 Let p = n-th odd prime. Then a(n) = least prime congruent to 5 modulo 8 such that Legendre(a(n), q) = -1 for all odd primes q <= p.

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A241482 Least fundamental discriminant D > 1 such that the first n primes p have (D/p) >= 0.

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