A268829 Square array A(row,col) = B(row,(2*col)-1), where B(p,q) = 0 if gcd(p,q) > 1, and 1 + 2*F(p,q) otherwise, where F is defined as in A269158. Array is read by descending antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

1, 1, 3, 1, 5, 3, 1, 15, 0, 1, 1, 9, 7, 1, 3, 1, 27, 7, 1, 5, 1, 1, 29, 0, 1, 0, 0, 3, 1, 23, 3, 1, 5, 9, 1, 3, 1, 17, 3, 1, 3, 15, 15, 5, 3, 1, 51, 0, 1, 3, 0, 0, 15, 0, 1, 1, 53, 7, 1, 13, 31, 11, 9, 1, 1, 3, 1, 63, 7, 1, 0, 21, 7, 27, 9, 0, 5, 3, 1, 57, 0, 1, 13, 0, 5, 29, 0, 13, 1, 0, 3, 1, 43, 3, 1, 3, 53, 15, 23, 9, 25, 1, 7, 1, 1

Offset: 1

Author

Antti Karttunen, Feb 20 2016

Keywords

• nonn
• tabl

Examples

			The top left [1 .. 16] x [1 .. 25] section of the array:
1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1
3,  5, 15,  9, 27, 29, 23, 17, 51, 53, 63, 57, 43, 45, 39, 33
3,  0,  7,  7,  0,  3,  3,  0,  7,  7,  0,  3,  3,  0,  7,  7
1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1
3,  5,  0,  5,  3,  3, 13,  0, 13,  3, 11, 13,  0, 13, 11, 11
1,  0,  9, 15,  0, 31, 21,  0, 53, 51,  0, 59, 41,  0, 33, 39
3,  1, 15,  0, 11,  7,  5, 15,  5,  3,  0,  7,  3,  9, 11,  9
3,  5, 15,  9, 27, 29, 23, 17, 51, 53, 63, 57, 43, 45, 39, 33
3,  0,  1,  9,  0,  9, 19,  0, 25,  3,  0,  1, 25,  0,  9, 19
1,  1,  0, 13, 25, 31, 27,  0, 63, 55, 53, 53,  0, 33, 45, 43
3,  5,  1,  1, 27,  0, 15, 23, 29, 27, 29,  7, 17, 21, 21, 31
3,  0,  7,  7,  0,  3,  3,  0,  7,  7,  0,  3,  3,  0,  7,  7
3,  1,  7, 15,  1, 29,  0, 13,  3, 23, 29, 17, 17, 19, 25, 23
1,  5,  1,  0, 17, 27, 19, 31, 55, 55,  0, 63, 41, 37, 45, 41
3,  0,  0,  1,  0,  1, 23,  0, 19,  7,  0, 31,  0,  0,  5, 31
1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1
3,  5, 15,  7, 27, 31,  1, 17,  0, 17, 35, 23, 17, 29, 37, 21
1,  0, 15,  1,  0, 21,  5,  0, 43, 55,  0, 57, 51,  0, 47, 51
3,  1,  1,  5,  1, 29, 21,  1, 51,  0, 23, 39, 17, 19, 21, 33
3,  5,  0,  5,  3,  3, 13,  0, 13,  3, 11, 13,  0, 13, 11, 11
3,  0,  1,  0,  0, 31, 23,  0,  1, 53,  0, 21, 35,  0, 21, 31
1,  1, 15,  9,  1,  0, 25,  7, 47, 47, 35, 63, 59, 57, 51, 63
3,  5,  7,  9,  3,  1, 27, 17, 53,  1, 63,  0, 27, 39, 17, 23
1,  0,  9, 15,  0, 31, 21,  0, 53, 51,  0, 59, 41,  0, 33, 39
3,  1,  0,  1, 11,  3,  3,  0, 51, 51,  1, 57,  0, 25, 51, 27
		

Crossrefs

Cf. arrays A268728, A269158.

Programs

Scheme

(define (A268829 n) (let ((p (A002260 n)) (q (+ -1 (* 2 (A004736 n))))) (if (< 1 (gcd p q)) 0 (+ 1 (* 2 (A269158auxbi p q)))))) ;; This one uses the code of A269158.
;; The following is a more stand-alone implementation:
(define (A268829 n) (A268829auxbi (A002260 n) (+ -1 (* 2 (A004736 n)))))
(define (A268829auxbi p q) (if (not (odd? q)) (error "A268829auxbi: the second argument should be odd: " p q) (let loop ((p p) (q q) (s 0)) (cond ((zero? p) 0) ((= 1 p) (+ 1 (* 2 s))) ((odd? p) (loop (modulo q p) p (A003987bi s (A004198bi p q)))) (else (loop (/ p 2) q (A003987bi s (A003987bi q (/ (- q 1) 2)))))))))

Formula

A(i,j) = B(i,(2*j)-1), where B(p,q) = 0 if gcd(p,q) > 1, and 1 + 2*F(p,q) = 1 + 2*A269158(p,(q+1)/2) otherwise, where function F is defined as in A269158.