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.

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

0,3

Comments

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.)

Links

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

Prog

(MIT Scheme:) (define (A166040 n) (let ((w (A005408 n))) (let loop ((i 1) (s 1) (zv 0)) (cond ((= i w) zv) ((zero? s) (loop (1+ i) (+ s (jacobi-symbol (1+ i) w)) (1+ zv))) (else (loop (1+ i) (+ s (jacobi-symbol (1+ i) w)) zv))))))

Crossrefs

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

Sequence in context: A101603 A228161 A124030 * A106378 A094301 A221542

Adjacent sequences: A166037 A166038 A166039 * A166041 A166042 A166043

Keyword

nonn

Author

Antti Karttunen, Oct 08 2009

Status

approved