A112049 -id:A112049 - cp's OEIS frontend

A112068 Positive integers i for which A112049(i) == 8.

180, 2855, 4199, 4619, 5195, 5399, 6719, 7475, 8555, 9000, 9815, 10739, 13859, 14340, 16235, 16319, 16524, 17159, 18299, 18564, 18744, 20015, 20579, 21359, 22320, 22524, 22619, 23759, 24275, 24504, 24960, 25884, 26544, 27455, 27720

Offset: 1

Antti Karttunen, Aug 27 2005

nonn

Row 8 of A112060.

A112069 Positive integers i for which A112049(i) == 9.

Original entry on oeis.org

264, 5279, 7895, 17135, 19979, 25475, 26615, 26940, 32579, 34799, 41435, 42900, 43320, 44039, 44459, 46019, 46284, 48959, 51000, 56939, 58560, 61595, 64259, 64595, 65364, 69155, 71135, 72084, 76295, 78624, 79379, 82320, 83159, 83304

Offset: 1

Antti Karttunen, Aug 27 2005

nonn

Row 9 of A112060.

A286579 Ordinal transform of A112049.

Original entry on oeis.org

1, 2, 1, 2, 3, 4, 3, 4, 5, 6, 1, 2, 7, 8, 5, 6, 9, 10, 7, 8, 11, 12, 3, 1, 13, 14, 9, 10, 15, 16, 11, 12, 17, 18, 2, 4, 19, 20, 13, 14, 21, 22, 15, 16, 23, 24, 5, 6, 25, 26, 17, 18, 27, 28, 19, 20, 29, 30, 3, 1, 31, 32, 21, 22, 33, 34, 23, 24, 35, 36, 7, 8, 37, 38, 25, 26, 39, 40, 27, 28, 41, 42, 9, 1, 43, 44, 29

Offset: 1

Antti Karttunen, May 26 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..32768

Cf. A112049.

A112049[n_] := For[i = 1, i <= 2n, i++, If[KroneckerSymbol[i, 2n+1] < 1, Return[PrimePi[i]]]];
b[_] = 0;
a[n_] := a[n] = With[{t = A112049[n]}, b[t] = b[t]+1];
Array[a, 105] (* Jean-François Alcover, Dec 21 2021 *)

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)).

A053761 Least positive integer k for which the Jacobi symbol (k|2n-1) is less than 1, where 2n-1 is a nonsquare; a(n)=0 if 2*n-1 is a square.

Original entry on oeis.org

0, 2, 2, 3, 0, 2, 2, 3, 3, 2, 2, 5, 0, 2, 2, 3, 3, 2, 2, 3, 3, 2, 2, 5, 0, 2, 2, 3, 3, 2, 2, 3, 3, 2, 2, 7, 5, 2, 2, 3, 0, 2, 2, 3, 3, 2, 2, 5, 5, 2, 2, 3, 3, 2, 2, 3, 3, 2, 2, 7, 0, 2, 2, 3, 3, 2, 2, 3, 3, 2, 2, 5, 5, 2, 2, 3, 3, 2, 2, 3, 3, 2, 2, 5, 0, 2, 2, 3, 3, 2, 2, 3, 3, 2, 2, 7, 5, 2, 2, 3

Offset: 1

Steven Finch, Apr 05 2000

nonn

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 443-448.
Paulo Ribenboim, The New Book of Prime Number Records, 3rd ed., Springer-Verlag 1996; Math. Rev. 96k:11112.

Antti Karttunen, Table of n, a(n) for n = 1..10000
R. Baillie and S. S. Wagstaff, Lucas pseudoprimes, Math. Comp. 35 (1980) 1391-1417; Math. Rev. 81j:10005
Steven R. Finch, Quadratic Residues [Broken link]
Steven R. Finch, Quadratic Residues [From the Wayback machine]

Cf. A010052, A053760, A112046, A112049, A268829.

A053761 := proc(n) if issqr(2*n-1) then return 0 ; else for k from 1 do if numtheory[jacobi](k,2*n-1) < 1 then return k; end if; end do: end if; end proc: seq(A053761(n),n=1..100) ; # R. J. Mathar, Aug 08 2010

a[n_] := If[IntegerQ[Sqrt[2*n - 1]], Return[0], For[ k = 1, True, k++, If[ JacobiSymbol[k, 2*n - 1] < 1 , Return[k]]]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jun 20 2013, after R. J. Mathar *)

A112046(n) = for(i=1,(2*n),if((kronecker(i,(n+n+1)) < 1),return(i)));
A053761(n) = if(issquare((2*n)-1),0,A112046(n-1));
for(n=1, 10000, write("b053761.txt", n, " ", A053761(n))); \\ Antti Karttunen, May 10 2017

(define (A053761 n) (if (= 1 n) 0 (* (- 1 (A010052 (+ n n -1))) (A112046 (- n 1))))) ;; Antti Karttunen, May 10 2017

a(1) = 0; for n > 1, a(n) = (1-A010052((2*n)-1)) * A112046(n-1). - Antti Karttunen, May 10 2017

More terms from R. J. Mathar, Aug 08 2010

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

Original entry on oeis.org

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)).

cp's OEIS Frontend

A112068 Positive integers i for which A112049(i) == 8.

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A112069 Positive integers i for which A112049(i) == 9.

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A286579 Ordinal transform of A112049.

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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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A053761 Least positive integer k for which the Jacobi symbol (k|2*n-1) is less than 1, where 2*n-1 is a nonsquare; a(n)=0 if 2*n-1 is a square.

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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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A053761 Least positive integer k for which the Jacobi symbol (k|2n-1) is less than 1, where 2n-1 is a nonsquare; a(n)=0 if 2*n-1 is a square.