A286465 -id:A286465 - cp's OEIS frontend

A286453 Compound filter: a(n) = P(A061395(n), A286465(n)), where P(n,k) is sequence A000027 used as a pairing function.

0, 2, 5, 11, 94, 5, 14, 254, 17, 9, 195, 47, 259, 500, 9, 11, 413, 138, 44, 303, 32, 20, 2784, 47, 354, 216, 5, 329, 506, 9, 77, 3161, 356, 35, 175, 107, 202, 2709, 216, 24, 11188, 14, 420, 356, 24, 285, 450, 498, 70, 2349, 35, 51, 115937, 5, 20, 329, 74, 310, 3420, 864, 1243, 336, 500, 11, 384, 20, 580, 47285, 87, 14, 615, 498, 1296, 3015, 9, 74, 3491, 216

Offset: 1

Antti Karttunen, May 14 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Indranil Ghosh, Python program to generate the sequence

Cf. A000027, A046523, A061395, A112049, A278223, A286356, A286372, A286452.

(define (A286453 n) (* (/ 1 2) (+ (expt (+ (A061395 n) (A286465 n)) 2) (- (A061395 n)) (- (* 3 (A286465 n))) 2)))

a(n) = (1/2)*(2 + ((A061395(n)+A286465(n))^2) - A061395(n) - 3*A286465(n)).

A112046 a(n) = the least k >= 1 for which the Jacobi symbol J(k,2n+1) is not +1 (thus is either 0 or -1).

Original entry on oeis.org

2, 2, 3, 3, 2, 2, 3, 3, 2, 2, 5, 5, 2, 2, 3, 3, 2, 2, 3, 3, 2, 2, 5, 7, 2, 2, 3, 3, 2, 2, 3, 3, 2, 2, 7, 5, 2, 2, 3, 3, 2, 2, 3, 3, 2, 2, 5, 5, 2, 2, 3, 3, 2, 2, 3, 3, 2, 2, 7, 11, 2, 2, 3, 3, 2, 2, 3, 3, 2, 2, 5, 5, 2, 2, 3, 3, 2, 2, 3, 3, 2, 2, 5, 13, 2, 2, 3, 3, 2, 2, 3, 3, 2, 2, 7, 5, 2, 2, 3, 3, 2, 2

Offset: 1

Antti Karttunen, Aug 27 2005

nonn

If we instead list the least k >= 1, for which Jacobi symbol J(k,2n+1) is 0, we get A090368.

It is easy to see that every term is prime. Because the Jacobi symbol is multiplicative as J(ab,m) = J(a,m)*J(b,m) and if for every index i>=1 and < x, J(i,m)=1, then if J(x,m) is 0 or -1, x cannot be composite (say y*z, with both y and z less than x), as then either J(y,m) or J(z,m) would be non-one, which contradicts our assumption that x is the first index where non-one value appears. Thus x must be prime.

Indranil Ghosh and A.H.M. Smeets, Table of n, a(n) for n = 1..20000 (first 1000 terms from Indranil Ghosh)

One more than A112050.
Bisections: A112047, A112048, and their difference: A112053.
Cf. A000040, A053760, A053761, A090368, A112049, A112060, A112070, A268829, A286465, A286466.

A112046(n) = for(i=1, (2*n), if((kronecker(i, (n+n+1)) < 1), return(i))); \\ Antti Karttunen, May 26 2017

from sympy import jacobi_symbol as J
def a(n):
    i=1
    while True:
        if J(i, 2*n + 1)!=1: return i
        else: i+=1
print([a(n) for n in range(1, 103)]) # Indranil Ghosh, May 11 2017

a(n) = A112050(n) + 1 = A000040(A112049(n)).

A112049 a(n) = position of A112046(n) in A000040.

Original entry on oeis.org

1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 3, 3, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 3, 4, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 4, 3, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 3, 3, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 4, 5, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 3, 3, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 3, 6, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 4, 3, 1, 1, 2, 2, 1, 1

Offset: 1

Antti Karttunen, Aug 27 2005

nonn

A112051 gives the first positions of distinct new values in this sequence, that seem also to be the positions of the first occurrence of each n, and thus the positions of the records. Compare also to A084921. - Antti Karttunen, May 26 2017

Indranil Ghosh (terms 1..1000) & Antti Karttunen, Table of n, a(n) for n = 1..32768

Cf. A049084, A112046, A112051, A112060, A165471, A268829, A286465, A286466.

Cf. A286579 (ordinal transform).

a112046[n_]:=Block[{i=1},While[JacobiSymbol[i, 2n + 1]==1, i++]; i];a049084[n_]:=If[PrimeQ[n], PrimePi[n], 0]; Table[a049084[a112046[n]], {n, 102}] (* Indranil Ghosh, May 11 2017 *)

A112049(n) = for(i=1, (2*n), if((kronecker(i, (n+n+1)) < 1), return(primepi(i)))); \\ Antti Karttunen, May 26 2017

from sympy import jacobi_symbol as J, isprime, primepi
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 a(n): return a049084(a112046(n))
print([a(n) for n in range(1, 103)]) # Indranil Ghosh, May 11 2017

a(n) = A049084(A112046(n)).

Unnecessary fallback-clause removed from the name by Antti Karttunen, May 26 2017

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

Original entry on oeis.org

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

Antti Karttunen, May 10 2017

nonn

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).

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

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

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)));

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

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

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

A286461 Compound filter (2-adic valuation of n & 4k+1,4k+3 prime-signature combination of 2n-1): a(n) = P(A001511(n), A286364((2*n)-1)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 5, 4, 9, 22, 5, 4, 32, 4, 5, 121, 9, 46, 437, 4, 20, 121, 17, 4, 24, 4, 5, 67, 14, 22, 17, 4, 24, 121, 5, 4, 2562, 211, 5, 121, 9, 4, 107, 121, 14, 7261, 5, 211, 24, 4, 17, 121, 41, 4, 2280, 4, 9, 254, 5, 4, 32, 4, 17, 67, 24, 22, 17, 631, 35, 121, 5, 121, 783, 4, 5, 121, 32, 211, 2280, 4, 9, 67, 17, 4, 41, 121, 5, 254, 9, 46, 2280, 4, 140, 121, 5, 4, 24

Offset: 1

Antti Karttunen, May 10 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pairing Function
Indranil Ghosh, Python program to generate the sequence

Cf. A000027, A001511, A046523, A278223, A286161, A286364, A286371, A286372, A286463, A286465.

(define (A286461 n) (* (/ 1 2) (+ (expt (+ (A001511 n) (A286364 (+ n n -1))) 2) (- (A001511 n)) (- (* 3 (A286364 (+ n n -1)))) 2)))

a(n) = (1/2)*(2 + ((A001511(n)+A286364((2*n)-1))^2) - A001511(n) - 3*A286364((2*n)-1)).

cp's OEIS Frontend

A286453 Compound filter: a(n) = P(A061395(n), A286465(n)), where P(n,k) is sequence A000027 used as a pairing function.

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A112046 a(n) = the least k >= 1 for which the Jacobi symbol J(k,2n+1) is not +1 (thus is either 0 or -1).

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A112049 a(n) = position of A112046(n) in A000040.

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

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Formula

A286461 Compound filter (2-adic valuation of n & 4k+1,4k+3 prime-signature combination of 2n-1): a(n) = P(A001511(n), A286364((2*n)-1)), where P(n,k) is sequence A000027 used as a pairing function.

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