A166040 -id:A166040 - cp's OEIS frontend

A166087 a(n) = Position of the first occurrence of n in A166040. -1 if it does not occur there.

0, 1, 2, 5, 6, 9, 10, 207, 14, 45, 30, 21, 52, 25, 26, 33, 124, 13, 66, 105, 22, 245, 70, 231, 50, 113, 110, 81, 118, 37, 86, 225, 49, 173, 142, 411, 98, 61, 58, 101, 198, 141, 134, 201, 148, 153, 334, 325, 158, 187, 103, 161, 178, 133, 190, 217, 292, 205, 242

Offset: 0

Antti Karttunen, Oct 08 2009

nonn

A166089 gives the corresponding odd numbers. See also A166097, A166098.

A166098 Distinct values of A166040 in the order of appearance.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 17, 8, 11, 20, 13, 14, 10, 15, 29, 9, 32, 24, 12, 38, 37, 74, 18, 22, 90, 27, 30, 62, 36, 39, 50, 19, 26, 25, 28, 161, 118, 16, 68, 53, 42, 84, 41, 34, 44, 45, 48, 51, 80, 97, 33, 52, 153, 49, 54, 89, 40, 43, 188, 57, 7, 98, 124, 55, 31, 125, 66, 23

Offset: 0

Antti Karttunen, Oct 08 2009

nonn

This is a permutation of nonnegative integers if all integers >= 0 occur in A166040 at least once. In that case A166099 gives the inverse permutation.

a(n) = A166040(A166097(n)). See also A166087.

A166097 Positions where A166040 obtains distinct new values.

Original entry on oeis.org

0, 1, 2, 5, 6, 9, 10, 13, 14, 21, 22, 25, 26, 30, 33, 37, 45, 49, 50, 52, 58, 61, 62, 66, 70, 73, 81, 86, 94, 98, 101, 103, 105, 110, 113, 118, 121, 122, 124, 130, 133, 134, 137, 141, 142, 148, 153, 158, 161, 166, 171, 173, 178, 181, 187, 190, 193, 198, 201, 202

Offset: 0

Antti Karttunen, Oct 08 2009

nonn

A166098 gives the values themselves. See also A166087.

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.

Original entry on oeis.org

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.

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.

A166406 a(n) = A166405(n)-A166100(n).

Original entry on oeis.org

-1, 1, 0, 7, -27, 11, 0, 30, 0, 19, 0, 69, -250, 9, 0, 93, 0, 70, 0, 156, 0, 43, 0, 235, -1029, 102, 0, 220, 0, 177, 0, 126, 0, 67, 0, 497, 0, 50, 0, 395, -2187, 249, 0, 522, 0, 182, 0, 760, 0, 0, 0, 515, 0, 321, 0, 888, 0, 230, 0, 1190, -6655, 246, 0, 635, 0, 655, 0

Offset: 0

Antti Karttunen, Oct 21 2009, Oct 22 2009

sign

Zeros occur at (A166409(k)-1)/2. The negative terms occur at positions given by A046092 (see the comment at A166040).

Sum of those positive i <= 2n+1, for which J(i,2n+1)=-1 minus sum of those positive i <= 2n+1, for which J(i,2n+1)=+1. Here J(i,k) is the Jacobi symbol.

Antti Karttunen, Table of n, a(n) for n = 0..65535

A125615(n)=a(A102781(n)). Cf. A166100, A166407-A166409. The cases where a(i)/A005408(i) is not integer seem also to be given by A166101.

from sympy import jacobi_symbol as J
def a(n):
    l=0
    m=0
    for i in range(1, 2*n + 2):
        if J(i, 2*n + 1)==-1: l+=i
        elif J(i, 2*n + 1)==1: m+=i
    return l - m
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 12 2017

A166085 Number of times Sum_{i=1..u} J(i,4n+1) obtains value zero when u ranges from 1 to (4n+1). Here J(i,k) is the Jacobi symbol.

Original entry on oeis.org

0, 2, 0, 4, 4, 6, 0, 8, 4, 8, 8, 20, 0, 14, 6, 10, 8, 14, 8, 20, 0, 10, 14, 20, 8, 24, 12, 12, 14, 38, 0, 74, 10, 18, 12, 22, 10, 22, 38, 22, 14, 18, 0, 30, 8, 20, 24, 62, 8, 36, 14, 14, 20, 36, 12, 26, 0, 18, 18, 28, 14, 118, 16, 22, 26, 68, 12, 42, 10, 24, 18, 34, 0, 36, 44

Offset: 0

Antti Karttunen, Oct 08 2009

nonn

All terms are even.

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

a(n) = A166040(A005843(n)). Bisection of A166040.

A166095 Bisection of A166089.

Original entry on oeis.org

1, 5, 13, 21, 29, 61, 105, 53, 249, 133, 45, 141, 101, 221, 237, 173, 99, 285, 197, 117, 397, 269, 297, 669, 317, 207, 357, 381, 585, 485, 1265, 189, 2297, 461, 261, 1597, 509, 125, 629, 797, 333, 1237, 275, 773, 2369, 147, 531, 789, 1433, 423, 1581, 1085

Offset: 0

Antti Karttunen, Oct 08 2009

nonn

a(n) = the least odd integer 2i+1, with A166040(i) = 2n.

a(n) = A166089(A005843(n)) = A005408(A166087(A005843(n))). Differs from A166090 for the first time at n=16. See also A166096.

A166090 a(n) = A016813(A166093(n)).

Original entry on oeis.org

1, 5, 13, 21, 29, 61, 105, 53, 249, 133, 45, 141, 101, 221, 237, 173, 521, 285, 197, 117, 397, 269, 297, 669, 317, 341, 357, 381, 585, 485, 1265, 189, 2297, 461, 261, 1597, 509, 125, 629, 797, 333, 1237, 325, 773, 2369, 941, 549, 789, 1433, 1109

Offset: 0

Antti Karttunen, Oct 08 2009

nonn

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

Differs from A166095 for the first time at n=16. See also A166096.

cp's OEIS Frontend

A166087 a(n) = Position of the first occurrence of n in A166040. -1 if it does not occur there.

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A166098 Distinct values of A166040 in the order of appearance.

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A166097 Positions where A166040 obtains distinct new values.

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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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A166096 Bisection of A166089. Leftmost column of A166092.

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A166406 a(n) = A166405(n)-A166100(n).

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Python

A166085 Number of times Sum_{i=1..u} J(i,4n+1) obtains value zero when u ranges from 1 to (4n+1). Here J(i,k) is the Jacobi symbol.

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A166095 Bisection of A166089.

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A166090 a(n) = A016813(A166093(n)).

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