A112060 -id:A112060 - 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).

A112061 Transpose of A112060.

Original entry on oeis.org

1, 3, 2, 11, 4, 5, 24, 12, 7, 6, 60, 35, 23, 8, 9, 84, 155, 59, 36, 15, 10, 144, 239, 275, 95, 47, 16, 13, 180, 779, 575, 335, 119, 48, 19, 14, 264, 2855, 1499, 659, 359, 120, 71, 20, 17, 420, 5279, 4199, 1535, 839, 419, 179, 72, 27, 18, 480, 9095, 7895, 4619

Offset: 1

Antti Karttunen, Aug 27 2005

Index entries for sequences that are permutations of the natural numbers

A112083 Column 2 of A112060.

Original entry on oeis.org

2, 4, 12, 35, 155, 239, 779, 2855, 5279, 9095, 15695, 59135, 350699, 183395, 1352339, 1477295, 1077959, 6922920, 3038555, 12705840, 14199120

Offset: 1

Antti Karttunen, Aug 28 2005

Also row 2 of A112061. Cf. A112084.

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

A112070 Square array A(x,y) = y-th odd number 2i+1 (i>=1) for which A112049(2i+1)=x, or 0 if no such i exists; read by descending antidiagonals.

Original entry on oeis.org

3, 5, 7, 11, 9, 23, 13, 15, 25, 49, 19, 17, 47, 71, 121, 21, 31, 73, 119, 311, 169, 27, 33, 95, 191, 551, 479, 289, 29, 39, 97, 239, 671, 1151, 1559, 361, 35, 41, 143, 241, 719, 1319, 2999, 5711, 529, 37, 55, 145, 359, 839, 1679, 3071, 8399, 10559, 841, 43, 57

Offset: 1

Antti Karttunen, Aug 27 2005

This is a permutation of odd numbers greater than unity provided that the sequence A112046 contains only prime values and every prime occurs infinitely many times there. Because the Jacobi symbol is multiplicative with respect to its modulus, it follows that if n occurs on row i and m occurs on row j, then n*m cannot occur before row min(i,j).

			The top left corner of the array:
3,5,11,13,19,21,...
7,9,15,17,31,33,...
23,25,47,73,95,...

A(x, y) = 2*A112060(x, y)+1. Transpose: A112071. Column 1: A112052. Row 1: A047621, Row 2: A112072 Row 3: A112073, Row 4: A112074, Row 5: A112075, Row 6: A112076, Row 7: A112077, Row 8: A112078, Row 9: A112079.

A166091 Square array A(row>=0, col>=0) = (A166092(row,col)-3)/4, listed antidiagonally as A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), ...

Original entry on oeis.org

0, 1, 2, 3, 79, 4, 5, 151, 8, 103, 7, 175, 14, 409, 22, 9, 223, 15, 589, 43, 10, 11, 265, 25, 998, 143, 31, 12, 13, 275, 32, 1297, 157, 73, 20, 16, 17, 283, 62, 1364, 182, 158, 55, 28, 6, 19, 361, 69, 1891, 293, 164, 183, 34, 26, 52, 21, 373, 74, 1952, 397, 401

Offset: 0

Antti Karttunen, Oct 08 2009

Note: This is not a permutation of nonnegative integers, as for some odd n, A166040(n) gets even value, the first example being A166040(49)=32, thus 24 (= (49-1)/2) is missing from here, and correspondingly, 99 (= 2*49 + 1) is missing from A166092. Sequence A165602 gives the natural numbers missing from this table.

			The top left corner of the array:
0, 1, 3, 5, 7, 9, ...
2, 79, 151, 175, 223, 265, ...
4, 8, 14, 15, 25, 32, ...
103, 409, 589, 998, 1297, 1364, ...
22, 43, 143, 157, 182, 293, ...

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

See A166092. The leftmost column: A166094. The first five rows: A165468, A166052, A166054, A166056, A166058. Cf. also A112060.

A112051 a(1)=1, a(n) = first index i (> a(n-1)), where A112046(i) gets a value distinct from any values A112046(1)..A112046(a(n-1)).

Original entry on oeis.org

1, 3, 11, 24, 60, 84, 144, 180, 264, 420, 480, 684, 840, 924, 1104, 1404, 1740, 1860, 2244, 2520, 2664, 3120, 3444, 3960, 4704, 5100, 5304, 5724, 5940, 6384, 8064, 8580, 9384, 9660, 11100, 11400, 12324, 13284, 13944, 14964, 16020, 16380, 18240

Offset: 1

Antti Karttunen, Aug 27 2005

nonn

Column 1 of A112060 (row 1 of A112061). Cf. A112052.

A112062 Positive integers i for which A112049(i) == 2.

Original entry on oeis.org

3, 4, 7, 8, 15, 16, 19, 20, 27, 28, 31, 32, 39, 40, 43, 44, 51, 52, 55, 56, 63, 64, 67, 68, 75, 76, 79, 80, 87, 88, 91, 92, 99, 100, 103, 104, 111, 112, 115, 116, 123, 124, 127, 128, 135, 136, 139, 140, 147, 148, 151, 152, 159, 160, 163, 164, 171, 172, 175, 176

Offset: 1

Antti Karttunen, Aug 27 2005

nonn

Row 2 of A112060.

A112063 Positive integers i for which A112049(i) == 3.

Original entry on oeis.org

11, 12, 23, 36, 47, 48, 71, 72, 83, 96, 107, 108, 131, 132, 143, 156, 167, 168, 191, 192, 203, 216, 227, 228, 251, 252, 263, 276, 287, 288, 311, 312, 323, 336, 347, 348, 371, 372, 383, 396, 407, 408, 431, 432, 443, 456, 467, 468, 491, 492, 503, 516, 527

Offset: 1

Antti Karttunen, Aug 27 2005

nonn

Row 3 of A112060.

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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A112061 Transpose of A112060.

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A112083 Column 2 of A112060.

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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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A112070 Square array A(x,y) = y-th odd number 2i+1 (i>=1) for which A112049(2i+1)=x, or 0 if no such i exists; read by descending antidiagonals.

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A166091 Square array A(row>=0, col>=0) = (A166092(row,col)-3)/4, listed antidiagonally as A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), ...

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A112051 a(1)=1, a(n) = first index i (> a(n-1)), where A112046(i) gets a value distinct from any values A112046(1)..A112046(a(n-1)).

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A112062 Positive integers i for which A112049(i) == 2.

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A112063 Positive integers i for which A112049(i) == 3.

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