A329307 Define b(D) = -Sum_{i=1..D} Kronecker(-D,i)*i for D == 0 or 3 (mod 4); sequence gives D such that b(D) = 0 and b(D/k^2) != 0 for k > 1, given that D/k^2 is an integer == 0 or 3 (mod 4).

28, 60, 72, 92, 99, 100, 124, 147, 156, 180, 188, 207, 220, 275, 284, 315, 316, 348, 380, 412, 423, 444, 475, 476, 504, 507, 508, 531, 572, 600, 604, 612, 636, 639, 668, 676, 732, 747, 764, 775, 796, 847, 855, 860, 892, 924, 931, 936, 956, 963, 968, 975, 980, 988, 1020

Offset

1,1

Comments

Note that {Kronecker(D,i)} is a Dirichlet character mod |D| if and only if D == 0, 1 (mod 4).

Primitive terms in A329306.

Numbers of the form d*p^2, where -d is a fundamental discriminant and Kronecker(-d,p) = 1 (i.e., the rational prime p decomposes in the quadratic number field with discriminant -d).

Links

Table of n, a(n) for n=1..55.

Example

60 is a term because 60 = 2^2 * 15 and -15 is a fundamental discriminant. Indeed, -Sum_{i=1..60} Kronecker(-60,i)*i = 0 and -Sum_{i=1..15} Kronecker(-15,i)*i != 0.
Although -Sum_{i=1..252} Kronecker(-252,i)*i = 0, 252 is not a term, because 252/3^2 = 28 and -Sum_{i=1..28} Kronecker(-28,i)*i = 0

Prog

(PARI) isA329307(n) = if(n%4==0||n%4==3, my(f=factor(n)); for(i=1, omega(n), my(p=f[i, 1], e=f[i, 2], m=n/p^e); if(e==2 && isfundamental(-m) && kronecker(-m, p)==1, return(1)))); 0

Crossrefs

Cf. A329648, A329306.

Sequence in context: A042566 A132769 A329306 * A336088 A255159 A071750

Adjacent sequences: A329304 A329305 A329306 * A329308 A329309 A329310

Keyword

nonn

Author

Jianing Song, Nov 30 2019

Status

approved