A112070 -id:A112070 - cp's OEIS frontend

A112086 a(n) = the period of the first differences of the n-th row of A112060 (or A112070), or 0 if that row does not have a periodic first difference.

2, 4, 6, 16, 72, 420, 3240

Offset: 1

Antti Karttunen, Aug 28 2005

These values have been computed empirically. An independent recomputation or a mathematical proof would be welcome. The initial terms factored: 2, 2*2, 2*3, 2*2*2*3*3, 2*2*7*3*5, 2*2*2*3*3*3*3*5, ...

These are the periods of A010684, A112132, A112133, A112134, A112135, A112136, A112137, etc. (Periods of A112138 & A112139 not computed yet.) If we sum the period length prefixes of these sequences, as Sum_{i=1..a(1)} A010684(i), Sum_{i=1..a(2)} A112132(i), Sum_{i=1..a(3)} A112133(i), etc., we get the sequence 4, 12, 60, 420, 4620, 60060, 1021020, ... (cf. A097250) and when doubled, it yields: 8, 24, 120, 840, 9240, 120120, 2042040, ... (cf. A066631 and A102476).

A112071 Transpose of A112070.

Original entry on oeis.org

3, 7, 5, 23, 9, 11, 49, 25, 15, 13, 121, 71, 47, 17, 19, 169, 311, 119, 73, 31, 21, 289, 479, 551, 191, 95, 33, 27, 361, 1559, 1151, 671, 239, 97, 39, 29, 529, 5711, 2999, 1319, 719, 241, 143, 41, 35, 841, 10559, 8399, 3071, 1679, 839, 359, 145, 55, 37

Offset: 1

Antti Karttunen, Aug 27 2005

Index entries for sequences that are permutations of the natural numbers

A112084 Column 2 of A112070.

Original entry on oeis.org

5, 9, 25, 71, 311, 479, 1559, 5711, 10559, 18191, 31391, 118271, 701399, 366791, 2704679, 2954591, 2155919, 13845841, 6077111, 25411681, 28398241

Offset: 1

Antti Karttunen, Aug 28 2005

Note the subsequence equal to the portion of A045535 / A062241 and also the non-monotone drops present, like the one from a(13)=701399 to a(14)=366791.

An independent recomputation with another software, e.g. Mathematica, would be welcome.

Row 2 of A112071. a(n) = A005408(A112083(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)).

A047621 Numbers that are congruent to {3, 5} mod 8.

Original entry on oeis.org

3, 5, 11, 13, 19, 21, 27, 29, 35, 37, 43, 45, 51, 53, 59, 61, 67, 69, 75, 77, 83, 85, 91, 93, 99, 101, 107, 109, 115, 117, 123, 125, 131, 133, 139, 141, 147, 149, 155, 157, 163, 165, 171, 173, 179, 181, 187, 189, 195, 197, 203, 205, 211, 213, 219, 221, 227, 229

Offset: 1

N. J. A. Sloane

Numbers k for which Jacobi symbol J(2,k) = -1, so 2 (as well as 2^k) is not a square mod k. - Antti Karttunen, Aug 27 2005, corrected by Jianing Song, Nov 05 2019, see also A329095.

Numbers n whose multiplicative order modulo 2^k is 2^(k - 2) for k >= 4. For k = 3, the numbers whose multiplicative order modulo 8 is 2 are in sequence A047484. - Jianing Song, Apr 29 2018

Muniru A Asiru, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Row 1 of A112070. Complement of A047522 relative to A005408. Primes in this sequence: A003629.
Subsequence of A329095.
Cf. A047522, A066507, A101464, A144981.

a:=[3];; for n in [2..60] do a[n]:=8*n-a[n-1]-8; od; a; # Muniru A Asiru, Dec 04 2018

a047621 n = a047621_list !! (n-1)
a047621_list = 3 : 5 : map (+ 8) a047621_list
-- Reinhard Zumkeller, Jul 05 2013

LinearRecurrence[{1, 1, -1}, {3, 5, 11}, 100] (* Jean-François Alcover, Jul 31 2018 *)

a(n) = 8*n - a(n-1) - 8 (with a(1) = 3). - Vincenzo Librandi, Aug 06 2010
G.f.: x*(3 + 2*x + 3*x^2) / ( (1 + x)*(x - 1)^2 ). - R. J. Mathar, Oct 08 2011
A089911(3*a(n)) = 10. - Reinhard Zumkeller, Jul 05 2013
a(n) = 8*floor((n - 1)/2) + 4 + (-1)^n. - Gary Detlefs, Dec 03 2018
From Franck Maminirina Ramaharo, Dec 03 2018: (Start)
a(n) = 4*n - 2 - (-1)^n.
E.g.f.: 3 - (2 - 4*x)*exp(x) - exp(-x). (End)
a(n + 2) = a(n) + 8. - David A. Corneth, Dec 03 2018
Sum_{n>=1} (-1)^(n+1)/a(n) = (sqrt(2)-1)*Pi/8. - Amiram Eldar, Dec 11 2021
From Amiram Eldar, Nov 22 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = sec(Pi/8) (1/A144981).
Product_{n>=1} (1 + (-1)^n/a(n)) = 2*sin(Pi/8) (A101464). (End)

A112060 Square array A(x,y) = y-th natural number k for which A112049(k)=x and 0 if no such k exists; read by antidiagonals A(1,1), A(2,1), A(1,2), A(3,1), A(2,2), ...

Original entry on oeis.org

1, 2, 3, 5, 4, 11, 6, 7, 12, 24, 9, 8, 23, 35, 60, 10, 15, 36, 59, 155, 84, 13, 16, 47, 95, 275, 239, 144, 14, 19, 48, 119, 335, 575, 779, 180, 17, 20, 71, 120, 359, 659, 1499, 2855, 264, 18, 27, 72, 179, 419, 839, 1535, 4199, 5279, 420, 21, 28, 83, 204, 504

Offset: 1

Antti Karttunen, Aug 27 2005

This is a permutation of natural numbers provided that the sequence A112046 contains only prime values [which is true] and every prime occurs infinitely many times there.

			The top left corner of the array:
   1,  2,  5,  6,  9, 10, ...
   3,  4,  7,  8, 15, 16, ...
  11, 12, 23, 36, 47, ...

Index entries for sequences that are permutations of the natural numbers

A112070(x, y) = 2*A(X, Y)+1. Transpose: A112061. Column 1: A112051. Row 1: A042963, Row 2: A112062, Row 3: A112063, Row 4: A112064, Row 5: A112065, Row 6: A112066, Row 7: A112067, Row 8: A112068, Row 9: A112069.

Cf. also A227196.

A166092 Integers (all of the form 4k+3) organized into an array based on the number of times Sum_{i=1..u} J(i,4k+3) obtains value zero when u ranges from 1 to (4k+3), where J(i,k) is the Jacobi symbol.

Original entry on oeis.org

3, 7, 11, 15, 319, 19, 23, 607, 35, 415, 31, 703, 59, 1639, 91, 39, 895, 63, 2359, 175, 43, 47, 1063, 103, 3995, 575, 127, 51, 55, 1103, 131, 5191, 631, 295, 83, 67, 71, 1135, 251, 5459, 731, 635, 223, 115, 27, 79, 1447, 279, 7567, 1175, 659, 735, 139

Offset: 0

Antti Karttunen, Oct 08 2009

Note: these are all of the form 4k+3, but still this is not permutation of A004767 (for the reason explained in A166091). Sequence A165603 gives the 4k+3 integers missing from this table.This square array A(row>=0, col>=0) is listed antidiagonally as A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), ...

			The top left corner of the array:
3, 7, 15, 23, 31, 39, ...
11, 319, 607, 703, 895, 1063, ...
19, 35, 59, 63, 103, 131, ...
415, 1639, 2359, 3995, 5191, 5459, ...
91, 175, 575, 631, 731, 1175, ...

A. Karttunen, Table of n, a(n) for n = 0..10010

a(n) = A004767(A166091(n)). The leftmost column: A166096. The first five rows: A165469, A166053, A166055, A166057, A166059. Cf. also A112070.

A112052 a(n) = 2*A112051(n)+1.

Original entry on oeis.org

3, 7, 23, 49, 121, 169, 289, 361, 529, 841, 961, 1369, 1681, 1849, 2209, 2809, 3481, 3721, 4489, 5041, 5329, 6241, 6889, 7921, 9409, 10201, 10609, 11449, 11881, 12769, 16129, 17161, 18769, 19321, 22201, 22801, 24649, 26569, 27889, 29929

Offset: 1

Antti Karttunen, Aug 27 2005

nonn

From n>=4 onward seems to be squares of primes (A001248).

Column 1 of A112070 (row 1 of A112071).

A005408(A112051(n))

A112079 Odd numbers k for which 23 is the smallest positive i with Jacobi symbol J(i,k) != 1.

Original entry on oeis.org

529, 10559, 15791, 34271, 39959, 50951, 53231, 53881, 65159, 69599, 82871, 85801, 86641, 88079, 88919, 92039, 92569, 97919, 102001, 113879, 117121, 123191, 128519, 129191, 130729, 138311, 142271, 144169, 152591, 157249, 158759, 164641, 166319, 166609, 167879

Offset: 1

Antti Karttunen, Aug 27 2005

nonn

Andy Huchala, Table of n, a(n) for n = 1..20000

Row 9 of A112070.

a(n) = 2*A112069(n) + 1.

ls = []
for c in range(1, 10^5, 2):
    for i in range(1, 24):
        if jacobi_symbol(i, c) <= 0:
            if i < 23:
                break
            else:
                ls.append(c)
print(ls) # Andy Huchala, Feb 11 2023

a(31)-a(35) from Andy Huchala, Feb 10 2023

Name simplified by Sean A. Irvine, Feb 25 2023

A112072 Odd numbers n for which 3 is the smallest i (>= 1) with Jacobi symbol J(i,n) getting either a value 0 or -1.

Original entry on oeis.org

7, 9, 15, 17, 31, 33, 39, 41, 55, 57, 63, 65, 79, 81, 87, 89, 103, 105, 111, 113, 127, 129, 135, 137, 151, 153, 159, 161, 175, 177, 183, 185, 199, 201, 207, 209, 223, 225, 231, 233, 247, 249, 255, 257, 271, 273, 279, 281, 295, 297, 303, 305, 319, 321, 327

Offset: 1

Antti Karttunen, Aug 27 2005

nonn

Row 2 of A112070. a(n) = 2*A112062(n)+1.

cp's OEIS Frontend

A112086 a(n) = the period of the first differences of the n-th row of A112060 (or A112070), or 0 if that row does not have a periodic first difference.

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A112071 Transpose of A112070.

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A112084 Column 2 of A112070.

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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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A047621 Numbers that are congruent to {3, 5} mod 8.

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A112060 Square array A(x,y) = y-th natural number k for which A112049(k)=x and 0 if no such k exists; read by antidiagonals A(1,1), A(2,1), A(1,2), A(3,1), A(2,2), ...

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A166092 Integers (all of the form 4k+3) organized into an array based on the number of times Sum_{i=1..u} J(i,4k+3) obtains value zero when u ranges from 1 to (4k+3), where J(i,k) is the Jacobi symbol.

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A112052 a(n) = 2*A112051(n)+1.

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A112079 Odd numbers k for which 23 is the smallest positive i with Jacobi symbol J(i,k) != 1.

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A112072 Odd numbers n for which 3 is the smallest i (>= 1) with Jacobi symbol J(i,n) getting either a value 0 or -1.

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