A166092 -id:A166092 - cp's OEIS frontend

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), ...

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.

A166096 Bisection of A166089. Leftmost column of A166092.

Original entry on oeis.org

3, 11, 19, 415, 91, 43, 51, 67, 27, 211, 491, 463, 227, 163, 75, 451, 347, 823, 123, 203, 283, 403, 307, 651, 375, 323, 267, 435, 411, 587, 667, 1099, 1251, 683, 515, 835, 2623, 827, 1183, 795, 483, 627, 1059, 707, 387, 987, 1635, 763, 343, 1907

Offset: 0

Antti Karttunen, Oct 08 2009

nonn

a(n) = the least integer i of the form 4k+3, with A166040(i) = 2n+1.

A. Karttunen, Table of n, a(n) for n = 0..1305

a(n) = A004767(A166094(n)) = A166089(A005408(n)) = A005408(A166087(A005408(n))). See also A166096.

A166040 Number of times Sum_{i=1..u} J(i,2n+1) obtains value zero when u ranges from 1 to (2n+1). Here J(i,k) is the Jacobi symbol.

Original entry on oeis.org

0, 1, 2, 1, 0, 3, 4, 1, 4, 5, 6, 1, 0, 17, 8, 1, 4, 5, 8, 1, 8, 11, 20, 1, 0, 13, 14, 1, 6, 5, 10, 5, 8, 15, 14, 1, 8, 29, 20, 1, 0, 13, 10, 1, 14, 9, 20, 1, 8, 32, 24, 5, 12, 17, 12, 1, 14, 15, 38, 1, 0, 37, 74, 11, 10, 5, 18, 17, 12, 15, 22, 1, 10, 90, 22, 1, 38, 17, 22, 1, 14, 27, 18

Offset: 0

Antti Karttunen, Oct 08 2009

nonn

A046092 gives the positions of zeros, as only with odd squares A016754(m) = A005408(A046092(m)) Jacobi symbols J(i,n) never obtain value -1, and thus their partial sum never descends back to zero. Even positions contain only even values, while odd positions contain odd values in all other positions, except even values in the positions given by A005408(A165602(i)), for i>=0.

Four bold conjectures by Antti Karttunen, Oct 08 2009: 1) All odd natural numbers occur. 2) Each of them occurs infinitely many times. 3) All even natural numbers occur. 4) Each even number > 0 occurs only finitely many times. (The last can be disputed. For example, 6 occurs four times among the first 400001 terms, at the positions 10, 28, 360, 215832.)

Antti Karttunen, Table of n, a(n) for n = 0..400000

Bisections: A166085, A166086. See also A166087, A165601, A166092.

A166053 Integers of the form 4n+3 for which Sum_{i=1..u} J(i,4n+3) obtains value zero exactly 3 times, when u ranges from 1 to (4n+3). Here J(i,k) is the Jacobi symbol.

Original entry on oeis.org

11, 319, 607, 703, 895, 1063, 1103, 1135, 1447, 1495, 2287, 2383, 2543, 3223, 3967, 4255, 4615, 4807, 5143, 5207, 5407, 5695, 5783, 5983, 6247, 6487, 6607, 6799, 6943, 7495, 7927, 8503, 9103, 9127, 9655, 10183, 10615, 10735, 12055, 13207

Offset: 1

Antti Karttunen, Oct 08 2009

nonn

Second row of A166092.

a(n) = A004767(A166052(n)).

A166055 Integers of the form 4n+3 for which Sum_{i=1..u} J(i,4n+3) obtains value zero exactly 5 times, when u ranges from 1 to (4n+3). Here J(i,k) is the Jacobi symbol.

Original entry on oeis.org

19, 35, 59, 63, 103, 131, 251, 279, 299, 371, 395, 419, 495, 711, 755, 899, 927, 943, 971, 1091, 1139, 1143, 1207, 1359, 1595, 1739, 1743, 1791, 1811, 1931, 1979, 2007, 2219, 2411, 2423, 2435, 2439, 2651, 2655, 2703, 2771, 2847, 2871, 2915, 2939

Offset: 1

Antti Karttunen, Oct 08 2009

nonn

Third row of A166092.

Cf. A004767, A166054, A166092.

a(n) = A004767(A166054(n)).

A166057 Integers of the form 4n+3 for which Sum_{i=1..u} J(i,4n+3) obtains value zero exactly 7 times, when u ranges from 1 to (4n+3). Here J(i,k) is the Jacobi symbol.

Original entry on oeis.org

415, 1639, 2359, 3995, 5191, 5459, 7567, 7811, 8315, 8459, 8479, 9611, 9895, 9943, 11159, 12271, 12731, 12851, 13835, 14447, 15155, 15251, 16091, 17891, 18143, 18319, 21059, 21755, 23055, 23411, 23435, 23815, 24395, 24535, 26143, 27179

Offset: 1

Antti Karttunen, Oct 08 2009

nonn

Fourth row of A166092.

a(n) = A004767(A166056(n)).

A166059 Integers of the form 4n+3 for which Sum_{i=1..u} J(i,4n+3) obtains value zero exactly 9 times, when u ranges from 1 to (4n+3). Here J(i,k) is the Jacobi symbol.

Original entry on oeis.org

91, 175, 575, 631, 731, 1175, 1591, 2175, 2575, 3059, 3199, 3295, 3575, 4175, 4575, 4695, 5939, 5959, 6143, 6575, 7175, 7259, 7383, 7575, 7615, 8175, 9175, 9199, 9491, 9575, 10135, 10175, 10407, 10551, 10607, 10811, 10955, 11175, 12575

Offset: 1

Antti Karttunen, Oct 08 2009

nonn

Fifth row of A166092.

a(n) = A004767(A166058(n)).

cp's OEIS Frontend

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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A166096 Bisection of A166089. Leftmost column of A166092.

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A166040 Number of times Sum_{i=1..u} J(i,2n+1) obtains value zero when u ranges from 1 to (2n+1). Here J(i,k) is the Jacobi symbol.

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A166053 Integers of the form 4n+3 for which Sum_{i=1..u} J(i,4n+3) obtains value zero exactly 3 times, when u ranges from 1 to (4n+3). Here J(i,k) is the Jacobi symbol.

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A166055 Integers of the form 4n+3 for which Sum_{i=1..u} J(i,4n+3) obtains value zero exactly 5 times, when u ranges from 1 to (4n+3). Here J(i,k) is the Jacobi symbol.

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A166057 Integers of the form 4n+3 for which Sum_{i=1..u} J(i,4n+3) obtains value zero exactly 7 times, when u ranges from 1 to (4n+3). Here J(i,k) is the Jacobi symbol.

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A166059 Integers of the form 4n+3 for which Sum_{i=1..u} J(i,4n+3) obtains value zero exactly 9 times, when u ranges from 1 to (4n+3). Here J(i,k) is the Jacobi symbol.

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