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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). 1
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified January 25 10:03 EST 2022. Contains 350565 sequences. (Running on oeis4.)