A165468 -id:A165468 - cp's OEIS frontend

A166086 Number of times Sum_{i=1..u} J(i,4n+3) obtains value zero when u ranges from 1 to (4n+3). Here J(i,k) is the Jacobi symbol.

1, 1, 3, 1, 5, 1, 17, 1, 5, 1, 11, 1, 13, 1, 5, 5, 15, 1, 29, 1, 13, 1, 9, 1, 32, 5, 17, 1, 15, 1, 37, 11, 5, 17, 15, 1, 90, 1, 17, 1, 27, 1, 29, 9, 17, 1, 37, 1, 15, 1, 39, 50, 19, 1, 37, 13, 25, 1, 25, 1, 161, 19, 5, 1, 17, 1, 53, 1, 84, 5, 41, 1, 29, 11, 5, 1, 45, 1, 62, 3, 51, 1, 19

Offset: 0

Antti Karttunen, Oct 08 2009

nonn

A. Karttunen, Table of n, a(n) for n = 0..65536

a(n) = A166040(A005408(n)). Bisection of A166040. A165468 gives the positions of 1's, and respectively, A166052, A166054, A166056 and A166058 give the positions of 3's, 5's, 7's and 9's in this sequence. Note how 3's seem to be more rare than 5's, and 7's more rare than 9's.

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.

A165469 Numbers of the form 4n+3 for which Sum_{i=1..u} J(i,4n+3) is never zero for any u in range [1,(2n+1)], where J(i,m) is the Jacobi symbol.

Original entry on oeis.org

3, 7, 15, 23, 31, 39, 47, 55, 71, 79, 87, 95, 111, 119, 143, 151, 159, 167, 183, 191, 199, 215, 231, 239, 255, 263, 271, 287, 303, 311, 327, 335, 359, 383, 391, 399, 407, 431, 439, 447, 455, 471, 479, 503, 511, 519, 527, 535, 551, 559, 591, 599, 615, 623

Offset: 1

Antti Karttunen, Oct 06 2009

nonn

a(n) = A004767(A165468(n)). Setwise difference of A095100 and A165608. A165580 gives the primes in this sequence.

A165607 a(n) = (A165608(n)-3)/4.

Original entry on oeis.org

2, 6, 8, 14, 15, 18, 20, 25, 32, 33, 42, 43, 60, 62, 69, 73, 74, 79, 85, 87, 90, 92, 93, 98, 103, 104, 123, 134, 140, 141, 143, 151, 158, 164, 168, 175, 177, 182, 183, 188, 195, 216, 218, 223, 224, 225, 231, 235, 242, 249, 265, 266, 272, 275, 281, 283, 284, 285

Offset: 1

Antti Karttunen, Oct 06 2009

nonn

Set-wise difference of A095274 and A165468.

cp's OEIS Frontend

A166086 Number of times Sum_{i=1..u} J(i,4n+3) obtains value zero when u ranges from 1 to (4n+3). Here J(i,k) is the Jacobi symbol.

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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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A165469 Numbers of the form 4n+3 for which Sum_{i=1..u} J(i,4n+3) is never zero for any u in range [1,(2n+1)], where J(i,m) is the Jacobi symbol.

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A165607 a(n) = (A165608(n)-3)/4.

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