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A276004
a(n) is the number of nonzero digits in the factorial-base representation of n that are matched by more significant digits from left; a(n) = A060502(n) - A060128(n).
6
0, 0, 0, 1, 0, 0, 0, 0, 1, 2, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 2, 3, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 0, 0, 1, 2, 1, 1, 0, 0, 1, 2, 1, 1, 0, 1, 1, 2, 1, 2, 0, 0, 1, 2, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 2, 1, 2, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 2, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0
OFFSET
0,10
COMMENTS
a(n) is the number of times a nonzero digit d_i appears in position i of the factorial-base representation of n (where the least significant digit is in position 1) such that there is another nonzero digit d_j in such position j > i that j - d_j = i.
FORMULA
a(n) = A060502(n) - A060128(n).
a(n) = A000120(2*A275727(n) AND A276010(n)), where AND is a bitwise-and given in A004198.
EXAMPLE
For n=15 ("211" in factorial base) the least significant 1 at position 1 is matched by its immediate left neighbor 1 and also by 2 at position 3, as (2-1) = (3-2) = 1, the position where the least significant 1 itself is. However, this is counted just as one match, because this sequence gives the number of digits that are matched, instead of the number of digits that match, thus a(15)=1.
PROG
(Scheme, two variants)
(define (A276004 n) (- (A060502 n) (A060128 n)))
(define (A276004 n) (let ((fv (list->vector (cons 0 (reverse (n->factbase n)))))) (let loop ((i 1) (c 0)) (if (>= i (vector-length fv)) c (let ((d (vector-ref fv i))) (cond ((zero? d) (loop (+ 1 i) c)) ((zero? (vector-ref fv (- i d))) (loop (+ 1 i) c)) (else (begin (vector-set! fv (- i d) 0) (loop (+ 1 i) (+ 1 c))))))))))
(define (n->factbase n) (let loop ((n n) (fex (if (zero? n) (list 0) (list))) (i 2)) (cond ((zero? n) fex) (else (loop (floor->exact (/ n i)) (cons (modulo n i) fex) (+ 1 i))))))
CROSSREFS
Cf. A276005 (indices of zeros), A276006 (of nonzeros).
Differs from A276007 for the first time at n=15, where a(15)=1, while A276004(15)=2.
Sequence in context: A287156 A092510 A117208 * A133300 A337101 A178779
KEYWORD
nonn,base
AUTHOR
Antti Karttunen, Aug 17 2016
STATUS
approved