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

Author

Leroy Quet, Jul 21 2006

Keywords

• nonn

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

Examples

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

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

Crossrefs

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

Programs

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 *)

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)))))

Extensions

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