A329306 - OEIS

%I #10 Dec 05 2019 04:20:35

%S 28,60,72,92,99,100,112,124,147,156,180,188,207,220,240,252,275,284,

%T 288,315,316,348,368,380,396,400,412,423,444,448,475,476,496,504,507,

%U 508,531,540,572,588,600,604,612,624,636,639,648,668,676,700,720,732,747,752,764

%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.

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

%C 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).

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

%C 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.

%e 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.

%e 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.

%o (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

%Y Cf. A329648, A329307.

%K nonn

%O 1,1

%A _Jianing Song_, Nov 30 2019