A094842
Class numbers associated with entries of A094841.
Original entry on oeis.org
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 22, 22, 79, 107, 107, 311, 432, 487, 665, 665, 692, 1044, 1044, 1044, 1044, 1536, 13909, 13909, 15204, 29351, 29351, 44332, 70877, 70877, 70877, 70877, 149460, 223574, 296475, 296475, 553436, 553436, 553436
Offset: 1
- 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.
Original entry on oeis.org
19, 43, 67, 163, 77683, 1333963, 2404147, 20950603, 36254563, 51599563, 96295483, 114148483, 269497867, 585811843, 52947440683, 71837718283, 229565917267, 575528148427, 1432817816347, 6778817202523, 16501779755323, 30059924764123, 110587910656507, 4311527414591923, 10472407114788067, 22261805373620443, 132958087830686827, 441899002218793387, 2278509757859388307, 5694230275645018963, 9828323860172600203
Offset: 1
- 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.
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, 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]
-
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
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
- 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.
-
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
A306218
Fundamental discriminant D < 0 with the least absolute value such that the first n primes p have (D/p) >= 0, negated.
Original entry on oeis.org
4, 8, 15, 20, 24, 231, 264, 831, 920, 1364, 1364, 9044, 67044, 67044, 67044, 67044, 268719, 268719, 3604695, 4588724, 5053620, 5053620, 5053620, 5053620, 60369855, 364461096, 532735220, 715236599, 1093026360, 2710139064, 2710139064, 3356929784, 3356929784
Offset: 1
(-231/2) = 1, (-231/3) = 0, (-231/5) = 1, (-231/7) = 0, (-231/11) = 0, (-231/13) = 1, so 2, 5, 13 decompose in Q[sqrt(-231)] and 3, 7, 11 ramify in Q[sqrt(-231)]. For other fundamental discriminants -231 < D < 0, at least one of 2, 3, 5, 7, 11, 13 is inert in the imaginary quadratic field with discriminant D, so a(6) = 231.
-
a(n) = my(i=1); while(!isfundamental(-i)||sum(j=1, n, kronecker(-i,prime(j))==-1)!=0, i++); i
Showing 1-5 of 5 results.
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