A115751 a(1)=1. a(n) = number of positive divisors of n which are not among the first (n-1) terms of the sequence.

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

Offset

1,4

Comments

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

Links

Antti Karttunen, Table of n, a(n) for n = 1..10001

Example

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.

Maple

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

Prog

(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)))))
;; Antti Karttunen, Oct 21 2017

Crossrefs

Cf. A088167.

Sequence in context: A348383 A369713 A295635 * A324888 A249145 A048684

Adjacent sequences: A115748 A115749 A115750 * A115752 A115753 A115754

Keyword

nonn

Author

Leroy Quet, Mar 28 2006

Extensions

More terms from Emeric Deutsch, Apr 01 2006

Status

approved