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

1,2

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

Formula

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

Prog

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

Crossrefs

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

Sequence in context: A341804 A109623 A317406 * A127532 A127530 A279347

Adjacent sequences: A286463 A286464 A286465 * A286467 A286468 A286469

Keyword

nonn

Author

Antti Karttunen, May 10 2017

Status

approved