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

Author

Jianing Song, Nov 30 2019

Keywords

• nonn

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).

Examples

			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
		

Crossrefs

Cf. A329648, A329306.

Programs

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