A286466 Compound filter: a(n) = P(A112049(n), A046523(n)), where P(n,k) is sequence A000027 used as a pairing function.

1, 2, 5, 12, 2, 16, 5, 38, 7, 16, 9, 94, 2, 16, 23, 138, 2, 67, 5, 80, 16, 16, 9, 355, 7, 16, 38, 80, 2, 436, 5, 530, 16, 16, 40, 706, 2, 16, 23, 302, 2, 436, 5, 80, 67, 16, 9, 1228, 7, 67, 23, 80, 2, 277, 23, 302, 16, 16, 14, 2021, 2, 16, 80, 2082, 16, 436, 5, 80, 16, 436, 9, 2704, 2, 16, 80, 80, 16, 436, 5, 1178, 121, 16, 9, 2086, 16, 16, 23, 302, 2, 1771

Offset: 1

Author

Antti Karttunen, May 10 2017

Keywords

• nonn

Comments

Here the information combined together to a(n) consists of A046523(n), giving essentially the prime signature of n, and the index of the first prime p >= 1 for which the Jacobi symbol J(p,2n+1) is not +1 (i.e. is either 0 or -1), the value which is returned by A112049(n).

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000027, A046523, A112049, A286258, A286465.

Programs

PARI

A112049(n) = for(i=1,(2*n),if((kronecker(i,(n+n+1)) < 1),return(primepi(i))));
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ This function from Charles R Greathouse IV, Aug 17 2011
A286466(n) = (1/2)*(2 + ((A112049(n)+A046523(n))^2) - A112049(n) - 3*A046523(n));
for(n=1, 10000, write("b286466.txt", n, " ", A286466(n)));

Python

from sympy import jacobi_symbol as J, factorint, isprime, primepi
def P(n):
    f = factorint(n)
    return sorted([f[i] for i in f])
def a046523(n):
    x=1
    while True:
        if P(n) == P(x): return x
        else: x+=1
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
def a049084(n): return primepi(n) if isprime(n) else 0
def a112046(n):
    i=1
    while True:
        if J(i, 2*n + 1)!=1: return i
        else: i+=1
def a112049(n): return a049084(a112046(n))
def a(n): return T(a112049(n), a046523(n)) # Indranil Ghosh, May 11 2017

Scheme

(define (A286466 n) (* (/ 1 2) (+ (expt (+ (A112049 n) (A046523 n)) 2) (- (A112049 n)) (- (* 3 (A046523 n))) 2)))

Formula

a(n) = (1/2)*(2 + ((A112049(n)+A046523(n))^2) - A112049(n) - 3*A046523(n)).