A329306 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.

28, 60, 72, 92, 99, 100, 112, 124, 147, 156, 180, 188, 207, 220, 240, 252, 275, 284, 288, 315, 316, 348, 368, 380, 396, 400, 412, 423, 444, 448, 475, 476, 496, 504, 507, 508, 531, 540, 572, 588, 600, 604, 612, 624, 636, 639, 648, 668, 676, 700, 720, 732, 747, 752, 764

Offset

1,1

Comments

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

From the comment in A329648, D is a term if and only if there exists a prime p being a factor of D such that if we write D = p^e * s, gcd(p,s) = 1, then e is even and Kronecker(-s,p) = 1; if p = 2, then s == 7 (mod 8).

If D is a term, then so is D*k^2 for any k. The "primitive" terms are given by A329307.

Let S be the set of the positive integers congruent to 0 or 3 mod 4, S_2 = {2^e * s: e is even, e > 0, s == 7 (mod 8)}, S_p = {p^e * s: e is even, e > 0, s is in S, Kronecker(-s,p) = 1} for odd primes p, then S_p has density 1/(2p*(p+1)) over S; for any x in S, "x is in S_2", "x is in S_3", "x is in S_5", ... are mutually independent. This sequence is Union_{prime p} S_p, so this sequence has density 1 - Product_{primes p} (1 - 1/(2p*(p+1))) ~ 0.156234 over S.

Links

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

Example

60 is a term because 60 = 2^2 * 15 and 15 == 7 (mod 8), so we have -Sum_{i=1..60} Kronecker(-60,i)*i = 0.
99 is a term because 99 = 3^2 * 11 and Kronecker(-11,3) = 1, so we have -Sum_{i=1..99} Kronecker(-99,i)*i = 0.

Prog

(PARI) isA329306(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) && if(p==2, m%8==7, kronecker(-m, p)==1), return(1)))); 0

Crossrefs

Cf. A329648, A329307.

Sequence in context: A219216 A042566 A132769 * A329307 A336088 A255159

Adjacent sequences: A329303 A329304 A329305 * A329307 A329308 A329309

Keyword

nonn

Author

Jianing Song, Nov 30 2019

Status

approved