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A179417
a(n) is the binary number (shown here in decimal) constructed from quadratic residues of 65537 in range [(n^2)+1,(n+1)^2] in such a way that quadratic residues are mapped to 1-bits, and non-quadratic residues (as well as the multiples of 65537) to 0-bits, with the lower end of range mapped to less significant, and the higher end of range to more significant bits.
3
1, 5, 24, 104, 279, 2001, 4131, 17453, 88826, 362532, 1655660, 6120642, 25376649, 128526482, 301370205, 1756488602, 8046359747, 30854867177, 73845140753, 488906501177, 2106640948770, 6573967883049, 29711211505300
OFFSET
0,2
COMMENTS
The binary width of terms are 1, 3, 5, 7, 9, ... i.e., the successive odd numbers, as their partial sums give the squares, 1, 4, 9, 16, ... at which points there certainly is always a quadratic residue, which thus gives the most significant bit for each number.
EXAMPLE
In the range [(2^2)+1, (2+1)^2] (i.e., [5,9]) we have A165471(5)=A165471(6)=A165471(7)=-1 and A165471(8)=A165471(9)=+1, i.e., there are quadratic non-residues at points 5, 6 and 7, and quadratic residues at 8 and 9, so we construct a binary number 11000, which is 24 in decimal, thus a(2)=24.
PROG
(MIT/GNU Scheme)
(define (A179417 n) (let ((ul (A005408 n))) (let loop ((i (A000290 n)) (j 0) (s 0)) (cond ((= j ul) s) ((= 0 (1+halved (A165471 (1+ i)))) (loop (1+ i) (1+ j) s)) (else (loop (1+ i) (1+ j) (+ s (expt 2 j))))))))
(define (1+halved n) (floor->exact (/ (1+ n) 2)))
CROSSREFS
Cf. A179418.
Compare to similar bit triangle illustrations given in A080070, A122229, A122232, A122235, A122239, A122242, A122245.
Sequence in context: A276139 A078820 A291395 * A181305 A046724 A272578
KEYWORD
nonn,base
AUTHOR
Antti Karttunen, Jul 27 2010
STATUS
approved