A120992 Number of integers in n-th run of squarefree positive integers.

3, 3, 2, 3, 1, 1, 3, 1, 3, 3, 3, 3, 2, 1, 1, 1, 3, 2, 3, 3, 2, 3, 2, 3, 1, 1, 3, 1, 3, 3, 3, 3, 2, 2, 1, 3, 2, 3, 3, 2, 1, 1, 2, 3, 1, 1, 3, 1, 2, 3, 3, 3, 2, 3, 1, 1, 3, 2, 3, 3, 3, 3, 2, 3, 1, 1, 3, 1, 2, 1, 1, 3, 3, 2, 3, 1, 1, 2, 2, 3, 3, 2, 1, 1, 2, 3, 1

Offset

1,1

Comments

The values 1, 2 and 3 occur 309008, 251134 and 439858 times, respectively, in the first 1000000 terms. - Rick L. Shepherd, Jul 25 2006

From Reinhard Zumkeller, Jan 20 2008: (Start)

1 <= a(n) <= 3.

A136742(n) = Product{k=0..a(n)} (A072284(n)+k).

A136743(n) = Sum_{k=0..a(n)} A001221(A072284(n)+k).

(End)

Also the lengths of runs in A243348, differences of the n-th squarefree number and n. - Antti Karttunen, Jun 06 2014

Links

A. Karttunen & R. Zumkeller (the first 1000 terms), Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Squarefree

Example

The runs of squarefree integers are as follows: (1,2,3), (5,6,7), (10,11), (13,14,15), (17), (19), (21,22,23),...

Maple

with(numtheory): a:=proc(n) if mobius(n)=0 then n else fi end: A:=[0, seq(a(n), n=1..500)]: b:=proc(n) if A[n]-A[n-1]>1 then A[n]-A[n-1]-1 else fi end: seq(b(n), n=2..nops(A)); # Emeric Deutsch, Jul 24 2006

Mathematica

t = {}; cnt = 0; Do[If[SquareFreeQ[n], cnt++, If[cnt > 0, AppendTo[t, cnt]; cnt = 0]], {n, 500}]; t (* T. D. Noe, Mar 19 2013 *)

Prog

(PARI) n=1; while(n<1000, c=0; while(issquarefree(n), n++; c++); print1(c, ", "); while(!issquarefree(n), n++)) \\ Rick L. Shepherd, Jul 25 2006
(Scheme, with Antti Karttunen's IntSeq-library)
(define (A120992 n) (if (= n 1) (Aincr_points_of_A243348 n) (- (Aincr_points_of_A243348 n) (Aincr_points_of_A243348 (- n 1)))))
;; Using these two auxiliary functions, not submitted separately:
(define Aincr_points_of_A243348 (COMPOSE -1+ (NONZERO-POS 1 1 Afirst_diffs_of_A243348)))
(define (Afirst_diffs_of_A243348 n) (if (< n 2) (- n 1) (- (A243348 n) (A243348 (- n 1)))))

Crossrefs

Cf. A005117, A076259, A013928, A013929, A053797, A053806, A120993, A243348.

Sequence in context: A292600 A014967 A210851 * A228483 A129979 A274709

Adjacent sequences: A120989 A120990 A120991 * A120993 A120994 A120995

Keyword

nonn

Author

Leroy Quet, Jul 21 2006

Extensions

More terms from Emeric Deutsch and Rick L. Shepherd, Jul 25 2006

Status

approved