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A286465 Compound filter: a(1) = 1, a(n) = P(A112049(n-1), A278223(n)), where P(n,k) is sequence A000027 used as a pairing function. 6
1, 2, 2, 5, 12, 2, 2, 23, 5, 2, 16, 9, 18, 29, 2, 5, 23, 16, 2, 23, 5, 2, 67, 9, 25, 16, 2, 23, 23, 2, 2, 80, 23, 2, 16, 14, 9, 67, 16, 5, 138, 2, 16, 23, 5, 16, 16, 31, 9, 67, 2, 5, 467, 2, 2, 23, 5, 16, 67, 40, 33, 16, 29, 5, 23, 2, 16, 302, 5, 2, 16, 31, 31, 67, 2, 5, 80, 16, 2, 23, 23, 2, 436, 9, 42, 67, 2, 80, 23, 2, 2, 23, 23, 16, 277, 14, 9, 436, 2, 5 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

After a(1) = 1, the information combined together to a(n) consists of A046523(2n-1), giving essentially the prime signature of 2n-1, 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-1).

LINKS

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

FORMULA

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

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

A286465(n) = if(1==n, n, (1/2)*(2 + ((A112049(n-1)+A046523((2*n)-1))^2) - A112049(n-1) - 3*A046523((2*n)-1)));

for(n=1, 10000, write("b286465.txt", n, " ", A286465(n)));

(Scheme) (define (A286465 n) (if (= 1 n) n (* (/ 1 2) (+ (expt (+ (A112049 (- n 1)) (A046523 (+ -1 n n))) 2) (- (A112049 (- n 1))) (- (* 3 (A046523 (+ -1 n 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 a278223(n): return a046523(2*n - 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 1 if n==1 else T(a112049(n - 1), a278223(n)) # Indranil Ghosh, May 11 2017

CROSSREFS

Cf. A000027, A046523, A112049, A278223, A286461, A286466.

Sequence in context: A152445 A077182 A230090 * A208258 A208872 A208581

Adjacent sequences:  A286462 A286463 A286464 * A286466 A286467 A286468

KEYWORD

nonn

AUTHOR

Antti Karttunen, May 10 2017

STATUS

approved

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Last modified May 15 01:43 EDT 2021. Contains 343909 sequences. (Running on oeis4.)