A112046 -id:A112046 - cp's OEIS frontend

A112056 Odd numbers of the form 4n-1 for which Jacobi-first-non-one(4n-1) differs from Jacobi-first-non-one(4n+1).

47, 71, 119, 167, 191, 287, 311, 359, 407, 431, 479, 527, 551, 647, 671, 719, 767, 791, 839, 887, 911, 959, 1007, 1031, 1127, 1151, 1199, 1247, 1271, 1319, 1367, 1391, 1487, 1511, 1559, 1607, 1631, 1679, 1727, 1751, 1799, 1847, 1871, 1967

Offset: 1

Antti Karttunen, Aug 27 2005

nonn

Here Jacobi-first-non-one(m) (for odd numbers m) is defined as the first value of i >= 1, for which Jacobi symbol J(i,m) is not +1 (i.e. is either 0 or -1).

Cf. A112054, A112057, A112058.

a112046[n_]:=Block[{i=1}, While[JacobiSymbol[i, 2n + 1]==1, i++]; i]; 4*Select[Range[1000], a112046[2#] - a112046[2# - 1] != 0 &] - 1 (* Indranil Ghosh, May 24 2017 *)

from sympy import jacobi_symbol as J
def a112046(n):
    i=1
    while True:
        if J(i, 2*n + 1)!=1: return i
        else: i+=1
def a(n): return a112046(2*n) - a112046(2*n - 1)
print([4*n - 1 for n in range(1, 1001) if a(n)!=0]) # Indranil Ghosh, May 24 2017

a(n) = 4*A112054(n)-1.

a(n) = A112057(n)-2 = A112058(n)-1.

A112058 Mean of A112056 and A112057.

Original entry on oeis.org

48, 72, 120, 168, 192, 288, 312, 360, 408, 432, 480, 528, 552, 648, 672, 720, 768, 792, 840, 888, 912, 960, 1008, 1032, 1128, 1152, 1200, 1248, 1272, 1320, 1368, 1392, 1488, 1512, 1560, 1608, 1632, 1680, 1728, 1752, 1800, 1848, 1872, 1968

Offset: 1

Antti Karttunen, Aug 27 2005

nonn

a(n) = A112056(n)+1 = A112057(n)-1 = (A112056(n)+A112057(n))/2.

a112046[n_]:=Block[{i=1}, While[JacobiSymbol[i, 2n + 1]==1, i++]; i]; 4*Select[Range[1000], a112046[2#] - a112046[2# - 1] != 0 &] (* Indranil Ghosh, May 24 2017 *)

from sympy import jacobi_symbol as J
def a112046(n):
    i=1
    while True:
        if J(i, 2*n + 1)!=1: return i
        else: i+=1
def a(n): return a112046(2*n) - a112046(2*n - 1)
print([4*n for n in range(1, 1001) if a(n)!=0]) # Indranil Ghosh, May 24 2017

a(n) = 4*A112054(n).

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)).

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

A112055 a(n) = A112054(n)/6.

Original entry on oeis.org

2, 3, 5, 7, 8, 12, 13, 15, 17, 18, 20, 22, 23, 27, 28, 30, 32, 33, 35, 37, 38, 40, 42, 43, 47, 48, 50, 52, 53, 55, 57, 58, 62, 63, 65, 67, 68, 70, 72, 73, 75, 77, 78, 82, 83, 85, 87, 88, 90, 92, 93, 97, 98, 100, 102, 103, 107, 108, 110, 112, 113, 117, 118, 120, 122

Offset: 1

Antti Karttunen, Aug 27 2005

nonn

Cf. A112054, A112082 (complement), A112085 (first differences).

a112046[n_]:=Block[{i=1}, While[JacobiSymbol[i, 2n + 1]==1, i++]; i]; Select[Range[1000], a112046[2#] - a112046[2# - 1] != 0 &]/6 (* Indranil Ghosh, May 25 2017 *)

from sympy import jacobi_symbol as J
def a112046(n):
    i=1
    while True:
        if J(i, 2*n + 1)!=1: return i
        else: i+=1
def a(n): return a112046(2*n) - a112046(2*n - 1)
print([n//6 for n in range(1, 201) if a(n)!=0]) # Indranil Ghosh, May 25 2017

A112057 Odd numbers of the form 4n+1 for which Jacobi-first-non-one(4n-1) differs from Jacobi-first-non-one(4n+1).

Original entry on oeis.org

49, 73, 121, 169, 193, 289, 313, 361, 409, 433, 481, 529, 553, 649, 673, 721, 769, 793, 841, 889, 913, 961, 1009, 1033, 1129, 1153, 1201, 1249, 1273, 1321, 1369, 1393, 1489, 1513, 1561, 1609, 1633, 1681, 1729, 1753, 1801, 1849, 1873, 1969

Offset: 1

Antti Karttunen, Aug 27 2005

nonn

The definition of Jacobi-first-non-one is given in A112056.

a(n) = A112056(n)+2 = A112058(n)+1.

a112046[n_]:=Block[{i=1}, While[JacobiSymbol[i, 2n + 1]==1, i++]; i]; 4*Select[Range[1000], a112046[2#] - a112046[2# - 1] != 0 &] + 1 (* Indranil Ghosh, May 25 2017 *)

jfno(n) = my(k = 1); while(kronecker(k, n) == 1, k++); k;
lista(nn) = {forstep(n=5, nn, 4, if (jfno(n-2) != jfno(n), print1(n, ", ")););} \\ Michel Marcus, Jan 30 2018

a(n) = 4*A112054(n) + 1.

A112059 Nonzero terms of A112053 and A112080.

Original entry on oeis.org

2, -2, 4, 8, -2, 12, -6, 12, 2, -2, -6, 18, -6, 2, -6, -4, 2, -2, 18, 2, -2, 24, 6, -2, 6, -8, 4, 2, -2, -6, 32, -6, 2, -6, -10, 2, -2, 28, 2, -2, 4, 38, -2, 6, -6, 4, 2, -2, -4, 42, -6, 2, -8, -4, 2, -2, 2, -2, 6, 8, -2, 48, -6, 4, 2, -2, -10, 6, -12, 2, -6, -4, 2, -2, 2, -2, 52, 8

Offset: 1

Antti Karttunen, Aug 27 2005

sign

Antti Karttunen, Table of n, a(n) for n = 1..4096

Cf. A112053, A112054, A112080.

from sympy import jacobi_symbol as J
def a112046(n):
    i=1
    while True:
        if J(i, 2*n + 1)!=1: return i
        else: i+=1
def a(n): return a112046(2*n) - a112046(2*n - 1)
print([a(n) for n in range(1, 201) if a(n)!=0]) # Indranil Ghosh, May 25 2017

a(n) = A112053(A112054(n)).

A227198 Odd terms in A227197.

Original entry on oeis.org

9, 15, 25, 33, 39, 49, 57, 63, 81, 87, 95, 105, 111, 119, 121, 129, 135, 145, 153, 159, 169, 177, 183, 201, 207, 215, 225, 231, 249, 255, 265, 273, 279, 289, 297, 303, 321, 327, 335, 345, 351, 361, 369, 375, 385, 393, 399, 417, 423, 441, 447, 455, 465, 471, 489

Offset: 1

Antti Karttunen, Jul 06 2013

nonn

Gives all terms n for which A090368((n+1)/2) = A112046((n-1)/2).

Contains all odd squares. What else?

Cf. A227196, A227197, A090368, A112046.

A227196(n) = for(k=1, n, if(kronecker(k, n)<1, return(k)))
for(n=2,1000,if((0==kronecker(A227196(n),n)&&1==(n%2)),print1(n,", ")))

A112050 Length of the longest prefix of 1's in the Jacobi-vector {J(2n+1,1),J(2n+1,2),...,J(2n+1,2n)}.

Original entry on oeis.org

1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 4, 4, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 4, 6, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 6, 4, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 4, 4, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 6, 10, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 4, 4, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 4, 12, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 6, 4, 1, 1, 2, 2, 1, 1

Offset: 1

Antti Karttunen, Aug 27 2005

nonn

Cf. A112046.

a112046[n_]:=Block[{i=1}, While[JacobiSymbol[i, 2n + 1]==1, i++]; i]; Table[a112046[n] - 1, {n, 102}] (* Indranil Ghosh, May 24 2017 *)

from sympy import jacobi_symbol as J
def a112046(n):
    i=1
    while True:
        if J(i, 2*n + 1)!=1: return i
        else: i+=1
def a(n): return a112046(n) - 1
print([a(n) for n in range(1, 103)]) # Indranil Ghosh, May 24 2017

a(n) = A112046(n) - 1.

Name clarified by Joerg Arndt, May 24 2017

A298991 Indices i in A112058 where records of 17i - 3A112058(i)/8 occur.

Original entry on oeis.org

2, 5, 13, 21, 24, 32, 40, 43, 51, 1470, 1478, 2701, 12032, 12040, 12048, 12051, 12059, 12067, 12070, 12078, 13301, 14524, 14683, 14691, 14699, 14702, 14710, 14718, 14721, 14729

Offset: 1

A.H.M. Smeets, Jan 31 2018

Cf. A112046, A112056, A112057, A112058.

i, n, rec = 0, 0, 0
while n < 1000:
    i = i+1
    if 17*i-3*A112058(i)//8 > rec:
        n, rec = n+1, 17*i-3*A112058(i)//8
        print(n,i)

cp's OEIS Frontend

A112056 Odd numbers of the form 4n-1 for which Jacobi-first-non-one(4n-1) differs from Jacobi-first-non-one(4n+1).

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A112058 Mean of A112056 and A112057.

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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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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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A112055 a(n) = A112054(n)/6.

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A112057 Odd numbers of the form 4n+1 for which Jacobi-first-non-one(4n-1) differs from Jacobi-first-non-one(4n+1).

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Keywords

Comments

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Programs

Mathematica

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A112059 Nonzero terms of A112053 and A112080.

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Python

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A227198 Odd terms in A227197.

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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.

A298991 Indices i in A112058 where records of 17i - 3A112058(i)/8 occur.