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A115751
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a(1)=1. a(n) = number of positive divisors of n which are not among the first (n-1) terms of the sequence.
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1
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1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 4, 1, 2, 3, 2, 1, 3, 1, 3, 2, 2, 1, 4, 2, 2, 2, 3, 1, 5, 1, 3, 2, 2, 2, 5, 1, 2, 2, 4, 1, 5, 1, 3, 3, 2, 1, 6, 2, 3, 2, 3, 1, 4, 2, 5, 2, 2, 1, 6, 1, 2, 4, 4, 2, 4, 1, 3, 2, 5, 1, 7, 1, 2, 3, 3, 2, 4, 1, 6, 3, 2, 1, 6, 2, 2, 2, 5, 1, 7, 2, 3, 2, 2, 2, 7, 1, 3, 4, 5, 1, 4, 1, 5, 4
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OFFSET
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1,4
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COMMENTS
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There are only 40 distinct values among the first 10000 terms. The records occur at positions: 1, 4, 12, 30, 48, 72, 120, 180, 240, 360, 480, 720, 840, 1260, 1680, 2160, 2520, 4620, 5040, ... - Antti Karttunen, Oct 21 2017
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LINKS
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EXAMPLE
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The divisors of 12 are 1, 2, 3, 4, 6 and 12. Of these, only the four divisors 3, 4, 6 and 12 do not occur among the first 11 terms of the sequence. So a(12) = 4.
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MAPLE
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with(numtheory): a[1]:=1: for n from 2 to 120 do div:=divisors(n): M:=convert([seq(a[j], j=1..n-1)], set): a[n]:=nops(div minus M): od: seq(a[n], n=1..120); # Emeric Deutsch, Apr 01 2006
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PROG
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(Scheme)
;; We define a mutual recurrence with the memoization-macro definec:
(definec (A115751 n) (if (= 1 n) n (length (remove (lambda (d) (zero? (modulo (Aauxseq_forA115751 (- n 1)) (A000040 d)))) (divisors n)))))
;; The other member of the mutual recurrence has not been submitted. Its n-th term keeps track in its prime factorization what distinct values has so far occurred in A115751(1) .. A115751(n). That is, iff value k has occurred in range a(1) .. a(n), then the n-th term of this auxiliary sequence is divisible by the k-th prime:
(definec (Aauxseq_forA115751 n) (if (= 1 n) 2 (lcm (A000040 (A115751 n)) (Aauxseq_forA115751 (- n 1)))))
(define (divisors n) (cons 1 (proper-divisors n)))
(define (proper-divisors n) (let loop ((k n) (divs (list))) (cond ((= 1 k) divs) ((zero? (modulo n k)) (loop (- k 1) (cons k divs))) (else (loop (- k 1) divs)))))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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