A329307 - OEIS

%I #7 Dec 05 2019 04:20:10

%S 28,60,72,92,99,100,124,147,156,180,188,207,220,275,284,315,316,348,

%T 380,412,423,444,475,476,504,507,508,531,572,600,604,612,636,639,668,

%U 676,732,747,764,775,796,847,855,860,892,924,931,936,956,963,968,975,980,988,1020

%N 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).

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

%C Primitive terms in A329306.

%C 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).

%e 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.

%e 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

%o (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

%Y Cf. A329648, A329306.

%K nonn

%O 1,1

%A _Jianing Song_, Nov 30 2019